% Jean-Pierre Demailly
% Universit\'e de Grenoble I, Institut Fourier
% \'Equations Diff\'erentielles et Analyse Num\'erique
% Collection Grenoble-Sciences, Presses Universitaires de Grenoble
% Chapitre VI

\input equa-mac.tex

\blankline\blankline
\hfill\chtitle Chapitre VI|%
\blankline\blankline
\hfill\title M\'ethodes de r\'esolution explicite|
\vskip6pt
\hfill\title des \'equations diff\'erentielles|
\chaptitlerunning VI -- M\'ethodes de r\'esolution explicite|
\vskip4pc

\noindent On se propose d'\'etudier un certain nombre de types
classiques d'\'equations diff\'erentielles du premier et du second
ordre pour lesquelles on sait ramener le calcul des solutions \`a des
calculs de primitives. Ceci fournira l'occasion d'illustrer les
r\'esultats g\'en\'eraux du chapitre V par des exemples.

\section 1. \'Equations du premier ordre|

\subsection 1.1. Remarques g\'en\'erales|

\noindent On consid\`ere une \'equation diff\'erentielle
$$
\frac{dy}{dx}=f(x,y)\leqno{\rm (E)}
$$
o\`u $f:U\rightarrow\bR$ est une fonction continue sur un ouvert 
$U\subset \bR^2$, localement lipschitzienne en $y$.

Les diff\'erentes solutions de l'\'equation (E) s'\'ecrivent en g\'en\'eral sous la forme
$$
y=\varphi(x,\lambda)
$$
o\`u $\lambda$ est un param\`etre r\'eel~: on dit parfois que la
solution \og$\,$g\'en\'erale$\,$\fg\ d\'epend d'un seul param\`etre. Pour
comprendre ce ph\'enom\`ene, il suffit d'appliquer le th\'eor\`eme de
Cauchy-Lipschitz~: si on cherche les solutions d\'efinies au voisinage
d'un point $x_0$, on sait qu'il existe une solution $y$ et une seule
telle que $y(x_0)=y_0$~; on peut donc choisir $\lambda=y_0$ pour
param\'etrer les solutions. Dans la pratique, le param\`etre $\lambda$
{\it appara\^{\i}t souvent comme constante d'int\'egration}.

Il arrive parfois qu'en plus de la solution g\'en\'erale on ait des
solutions particuli\`eres $y=\psi_0(x)$, $y=\psi_1(x)$, $\dots$ qui ne
s'obtiennent pour aucune valeur de $\lambda$~: on dit que ce sont des
{\it solutions singuli\`eres} (ou {\it courbes int\'egrales
singuli\`eres}) de (E).

On va maintenant d\'ecrire une situation un peu plus g\'en\'erale qui
se ram\`ene au cas d'une \'equation du type consid\'er\'e ci-dessus.

\medskip
\state Syst\`emes diff\'erentiels autonomes dans un ouvert $U\subset
\bR^2$ -- | On suppose donn\'e un champ de vecteurs dans $U$,
c'est-\`a-dire une application continue
$$
M\pmatrix{x\cr y\cr}\mapsto\overrightarrow V(M)\pmatrix{
a(x,y)\cr
b(x,y)\cr
},\quad M\in U.
$$
On appelle syst\`eme autonome associ\'e au champ de vecteurs
$\overrightarrow V(M)$ le syst\`eme diff\'erentiel
$$
\frac{d\overrightarrow M}{dt}=\overrightarrow V(M)\Leftrightarrow
\left\{\eqalign{
\dfrac{dx}{dt}=a(x,y)\hfill\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}=b(x,y)\hfill\cr
}.\right.\leqno{\rm (S)}
$$
Si $\overrightarrow V(M)$ repr\'esente un champ de vecteurs vitesse
(associ\'e par exemple \`a l'\'ecou\-le\-ment d'une nappe de fluide sur
une surface plane), r\'esoudre (S) revient \`a chercher la trajectoire
et la loi du mouvement des particules de fluide en fonction du temps.
Le mot \og$\,$autonome$\,$\fg\ signifie que le champ de vecteurs ne d\'epend
pas du temps $t$ (cas d'un \'ecoulement stationnaire).

Si $t\mapsto M(t)$ est solution, toute fonction $t\mapsto M(t+T)$
obtenue par un d\'ecalage dans le temps est encore solution. Dans
l'ouvert $U'=\big\{M(x,y)\,;\,\,a(x,y)\not=0\big\}$ on~a
${\rm (S)}\Rightarrow {\rm (E)}$ o\`u
$$
\frac{dy}{dx}=\frac{b(x,y)}{a(x,y)}=f(x,y).
\leqno{\rm (E)}
$$
R\'esoudre (E) permet de trouver la trajectoire des particules (mais pas la
loi du mouvement en fonction du temps).

\subsection 1.2. \'Equations \`a variables s\'epar\'ees|

\noindent Ce sont les \'equations dans lesquelles on peut regrouper
$x,dx$ d'une part et $y,dy$ d'autre part. Nous allons examiner 3 cas.

\bigskip
\noindent {\tenpointbf a) \'Equations $y'=f(x)$, avec $f:I\rightarrow\bR$
continue.} \medskip

Les solutions sont donn\'ees par
$$
y(x)=F(x)+\lambda,\quad\lambda\in\bR,
$$
o\`u $F$ est une primitive de $f$ sur $I$. Les courbes int\'egrales se
d\'eduisent les unes des autres par translations dans la direction $Oy$.
\bigskip

\vbox{
\noindent {\tenpointbf b) \'Equations $y'=g(y)$, avec $g:J\rightarrow\bR$
continue.}
\medskip

L'\'equation peut se r\'ecrire $\dfrac{dy}{dx}=g(y)$, ou encore
$\dfrac{dy}{g(y)}=dx$ \`a condition que $g(y)\not=0$.}

{\parindent=0.8pc
\item{$\bullet$} Notons $y_j$ les racines de $g(y)=0$ dans
l'intervalle $J$. Alors $y(x)=y_j$ est une solution (singuli\`ere)
\'evidente de l'\'equation.\par}

\noindent $\bullet$ Dans l'ouvert $U=\{(x,y)\in\bR\times J\,;\
g(y)\not=0\}$, on~a 
$$
{\rm (E)}\Leftrightarrow\frac{dy}{g(y)}=dx.
$$
Les solutions sont donn\'ees par
$$
G(y)=x+\lambda,\quad \lambda\in\bR
$$
o\`u $G$ est une primitive quelconque de $\dfrac{1}{g}$ sur chacun des
intervalles ouverts $[y_j,y_{j+1}[$ d\'elimit\'es par les racines de
$g$. Dans chaque bande $\bR\times{}]y_j,y_{j+1}[$, les courbes
int\'egrales se d\'eduisent les unes des autres par translations dans
la direction $Ox$~; ceci est \`a relier au fait que les lignes isoclines
sont les droites $y=m=$ constante.

Comme $G'=\dfrac{1}{g}$ et que $g$ est de signe constant sur
$]y_j,y_{j+1}[$, on en d\'eduit que $G$ est une application strictement
monotone bijective
$$
G:\ ]y_j,y_{j+1}[\ \rightarrow\ ]a_j,b_j[
$$
avec $a_j\in[-\infty,+\infty[$, $b_j\in{}]-\infty,+\infty]$. On peut
donc (au moins th\'eoriquement) exprimer $y$ en fonction de $x$~:
$$
y=G^{-1}(x+\lambda),\quad \lambda\in\bR.
$$
Supposons par exemple $g>0$, et par suite $G$ croissante sur $]y_j,y_{j+1}[$.
\medskip

\noindent$\bullet$ Si
$\int^{y_j+\varepsilon}_{y_j}\dfrac{dy}{g(y)}$ diverge, on
a $a_j=-\infty$, par cons\'equent $x=G(y)-\lambda\rightarrow-\infty$
quand $y\rightarrow y_j+0$. Dans ce cas, la courbe est asymptote \`a la
droite $y=y_j$.
\medskip
\noindent$\bullet$ Si $\int^{y_j+\varepsilon}_{y_j}\dfrac{dy}{g(y)}$ 
converge, alors $a_j\in\bR$ et $x\rightarrow a_j-\lambda$ quand $y\rightarrow
y_j+0$, avec de plus $y'=g(y)\rightarrow 0$~; la courbe vient rejoindre
la droite $y=y_j$ au point $(a_j-\lambda,y_j)$ et admet la droite
$y=y_j$ pour tangente en ce point. Cette situation montre qu'{\it il
n'y a pas unicit\'e} du probl\`eme de Cauchy en cas de convergence
de l'int\'egrale.
\bigskip

\majdeb
\state Exercice -- |  {\it V\'erifier que
$\int\dfrac{dy}{g(y)}$ est bien toujours divergente en tout
point $y_j$ tel que $g(y_j)=0$, lorsque $g$ est localement lipschitzienne.}
\par\majfin\par

\bigskip
L'allure des courbes int\'egrales est la suivante (dans le sch\'ema
ci-dessous, on suppose qu'il y a convergence en $y_2-0$, divergence
en $y_1\pm 0$ et $y_2+0$)~:

\InsertFig 16.000 87.000 {
1 mm unit 
  0.000  35.000 moveto 
 80.000   0.000 vector 
 30.000   0.000 moveto 
 80.000  90.000 vector 
  0.000  20.000 moveto 
  0.340 setlinewidth 
 80.000  20.000 lineto 
stroke 
  0.000  55.000 moveto 
 80.000  55.000 lineto 
stroke 
  0.113 setlinewidth
 83.500  68.000 moveto 
 12.000  90.000 vector 
 12.500 -90.000 vector 
 83.500  48.000 moveto 
  6.500  90.000 vector 
 27.500 -90.000 vector 
 83.500   8.000 moveto 
 11.500  90.000 vector 
  8.000 -90.000 vector 
gsave 
  0.000   0.000 moveto 
 74.000  80.000 rect  clip  newpath 
 46.000  10.000 moveto 
[  46.000  10.000    44.115  10.371    42.273  10.721    40.430  11.058   
 38.587  11.383    36.744  11.696    34.902  11.997    33.059  12.288   
 31.216  12.568    29.373  12.838    27.531  13.099    25.688  13.350   
 23.845  13.591    22.002  13.824    20.160  14.049    18.317  14.265   
 16.474  14.474    14.631  14.675    12.789  14.869    10.946  15.056   
  9.103  15.236     7.261  15.409     5.418  15.576     3.575  15.738   
  1.732  15.893    -0.110  16.043   ] scurve 
 46.000  10.000 moveto 
[  46.000  10.000    47.885   9.617    49.727   9.229    51.570   8.826   
 53.413   8.408    55.256   7.974    57.098   7.523    58.941   7.056   
 60.784   6.572    62.627   6.069    64.469   5.547    66.312   5.006   
 68.155   4.444    69.998   3.862    71.840   3.257    73.683   2.630   
 75.526   1.980   ] scurve 
 70.000  12.164 moveto 
[  70.000  12.164    68.045  12.466    66.133  12.750    64.222  13.023   
 62.310  13.286    60.398  13.539    58.486  13.782    56.575  14.017   
 54.663  14.243    52.751  14.460    50.840  14.669    48.928  14.870   
 47.016  15.064    45.105  15.250    43.193  15.429    41.281  15.602   
 39.370  15.768    37.458  15.928    35.546  16.082    33.635  16.230   
 31.723  16.373    29.811  16.510    27.900  16.642    25.988  16.769   
 24.076  16.892    22.165  17.009    20.253  17.123    18.341  17.232   
 16.430  17.337    14.518  17.438    12.606  17.535    10.694  17.629   
  8.783  17.719     6.871  17.806     4.959  17.889     3.048  17.969   
  1.136  18.047    -0.776  18.121   ] scurve 
 70.000  12.164 moveto 
[  70.000  12.164    71.955  11.853    73.867  11.537    75.778  11.208   
] scurve 
 46.843   1.428 moveto 
[  46.843   1.428    44.828   2.164    42.812   2.870    40.797   3.549   
 38.781   4.200    36.766   4.826    34.750   5.427    32.735   6.004   
 30.719   6.559    28.704   7.091    26.688   7.603    24.673   8.094   
 22.657   8.566    20.642   9.019    18.626   9.455    16.611   9.873   
 14.595  10.274    12.580  10.660    10.564  11.030     8.549  11.386   
  6.533  11.728     4.518  12.056     2.502  12.371     0.487  12.674   
 -1.528  12.965   ] scurve 
 46.843   1.428 moveto 
[  46.843   1.428    48.859   0.666    50.874  -0.127    52.890  -0.953   
] scurve 
  8.569   5.713 moveto 
[   8.569   5.713     6.419   6.316     4.269   6.894     2.119   7.447   
 -0.030   7.977   ] scurve 
  8.569   5.713 moveto 
[   8.569   5.713    10.719   5.087    12.869   4.434    15.018   3.752   
 17.168   3.040    19.318   2.297    21.468   1.521    23.618   0.711   
 25.767  -0.135   ] scurve 
 71.407  17.138 moveto 
[  71.407  17.138    69.268  17.259    67.177  17.373    65.085  17.482   
 62.994  17.587    60.902  17.687    58.810  17.783    56.719  17.875   
 54.627  17.964    52.536  18.048    50.444  18.130    48.353  18.208   
 46.261  18.282    44.170  18.354    42.078  18.423    39.986  18.489   
 37.895  18.552    35.803  18.613    33.712  18.671    31.620  18.727   
 29.529  18.780    27.437  18.832    25.345  18.881    23.254  18.928   
 21.162  18.973    19.071  19.017    16.979  19.058    14.888  19.098   
 12.796  19.136    10.705  19.173     8.613  19.208     6.521  19.242   
  4.430  19.274     2.338  19.306     0.247  19.335    -1.845  19.364   
] scurve 
 71.407  17.138 moveto 
[  71.407  17.138    73.546  17.014    75.638  16.888   ] scurve 
  0.000   0.000 moveto 
 15.000  30.000 moveto 
[  15.000  30.000    13.381  29.336    11.761  28.709    10.142  28.118   
  8.523  27.561     6.933  27.047     5.638  26.651     4.342  26.274   
  3.047  25.917     1.752  25.578     0.456  25.257    -0.839  24.954   
] scurve 
 15.000  30.000 moveto 
[  15.000  30.000    16.619  30.705    18.239  31.448    19.858  32.229   
 21.477  33.048    23.096  33.905    24.716  34.798    26.335  35.727   
 27.954  36.689    29.573  37.682    31.193  38.703    32.812  39.749   
 34.431  40.814    36.050  41.893    37.670  42.981    39.289  44.071   
 40.908  45.156    42.527  46.228    44.147  47.279    45.766  48.298   
 47.385  49.278    49.004  50.209    50.624  51.081    51.949  51.745   
 53.008  52.240    53.913  52.635    54.712  52.963    55.375  53.218   
 56.002  53.445    56.533  53.627    57.044  53.791    57.468  53.921   
 57.893  54.043    58.256  54.142    58.595  54.230    58.935  54.314   
 59.213  54.379    59.484  54.439    59.756  54.496    59.988  54.543   
 60.206  54.585    60.423  54.624    60.632  54.661    60.806  54.690   
 60.980  54.717    61.154  54.744    61.312  54.766    61.451  54.786   
 61.590  54.804    61.729  54.821    61.861  54.837    61.972  54.850   
 62.083  54.862    62.195  54.874    62.306  54.885    62.397  54.894   
 62.486  54.903    62.575  54.910    62.664  54.918    62.748  54.925   
 62.819  54.930    62.891  54.936    62.962  54.941    63.033  54.946   
 63.098  54.950    63.155  54.954    63.212  54.957    63.269  54.961   
 63.326  54.964    63.377  54.967    63.423  54.969    63.469  54.971   
 63.514  54.973    63.560  54.976    63.603  54.977   ] scurve 
 20.000  25.000 moveto 
[  20.000  25.000    18.780  24.726    17.643  24.484    16.507  24.253   
 15.371  24.034    14.235  23.825    13.098  23.626    11.962  23.437   
 10.826  23.258     9.690  23.087     8.718  22.948     7.809  22.824   
  6.900  22.704     5.991  22.589     5.082  22.479     4.173  22.374   
  3.264  22.273     2.355  22.176     1.446  22.083     0.537  21.994   
 -0.372  21.909   ] scurve 
 20.000  25.000 moveto 
[  20.000  25.000    21.220  25.292    22.357  25.576    23.493  25.875   
 24.629  26.188    25.765  26.516    26.902  26.858    28.038  27.217   
 29.174  27.591    30.310  27.982    31.447  28.390    32.583  28.816   
 33.719  29.259    34.856  29.720    35.992  30.199    37.128  30.697   
 38.264  31.214    39.401  31.749    40.537  32.303    41.673  32.876   
 42.810  33.468    43.946  34.078    45.082  34.706    46.218  35.351   
 47.355  36.013    48.491  36.692    49.627  37.386    50.764  38.094   
 51.900  38.815    53.036  39.548    54.172  40.291    55.309  41.042   
 56.445  41.800    57.581  42.563    58.718  43.328    59.854  44.093   
 60.990  44.855    62.126  45.612    63.263  46.361    64.399  47.099   
 65.535  47.822    66.672  48.528    67.808  49.214    68.944  49.875   
 70.080  50.510    71.217  51.114    72.353  51.684    73.469  52.209   
 74.378  52.608   ] scurve 
 45.000  25.000 moveto 
[  45.000  25.000    43.780  24.726    42.643  24.484    41.507  24.253   
 40.371  24.034    39.235  23.825    38.098  23.626    36.962  23.437   
 35.826  23.258    34.690  23.087    33.718  22.948    32.809  22.824   
 31.900  22.704    30.991  22.589    30.082  22.479    29.173  22.374   
 28.264  22.273    27.355  22.176    26.446  22.083    25.537  21.994   
 24.628  21.909    23.719  21.827    22.810  21.749    21.901  21.674   
 20.992  21.603    20.083  21.534    19.174  21.468    18.265  21.405   
 17.356  21.345    16.447  21.288    15.538  21.232    14.629  21.180   
 13.720  21.129    12.811  21.081    11.902  21.034    10.993  20.990   
 10.084  20.948     9.175  20.907     8.266  20.868     7.357  20.831   
  6.448  20.796     5.539  20.762     4.630  20.729     3.721  20.698   
  2.812  20.669     1.903  20.640     0.993  20.613     0.084  20.587   
 -0.825  20.562   ] scurve 
 45.000  25.000 moveto 
[  45.000  25.000    46.220  25.292    47.357  25.576    48.493  25.875   
 49.629  26.188    50.765  26.516    51.902  26.858    53.038  27.217   
 54.174  27.591    55.310  27.982    56.447  28.390    57.583  28.816   
 58.719  29.259    59.856  29.720    60.992  30.199    62.128  30.697   
 63.264  31.214    64.401  31.749    65.537  32.303    66.673  32.876   
 67.810  33.468    68.946  34.078    70.082  34.706    71.218  35.351   
 72.355  36.013    73.491  36.692    74.627  37.386   ] scurve 
 70.000  25.000 moveto 
[  70.000  25.000    68.780  24.726    67.643  24.484    66.507  24.253   
 65.371  24.034    64.235  23.825    63.098  23.626    61.962  23.437   
 60.826  23.258    59.690  23.087    58.718  22.948    57.809  22.824   
 56.900  22.704    55.991  22.589    55.082  22.479    54.173  22.374   
 53.264  22.273    52.355  22.176    51.446  22.083    50.537  21.994   
 49.628  21.909    48.719  21.827    47.810  21.749    46.901  21.674   
 45.992  21.603    45.083  21.534    44.174  21.468    43.265  21.405   
 42.356  21.345    41.447  21.288    40.538  21.232    39.629  21.180   
 38.720  21.129    37.811  21.081    36.902  21.034    35.993  20.990   
 35.084  20.948    34.175  20.907    33.266  20.868    32.357  20.831   
 31.448  20.796    30.539  20.762    29.630  20.729    28.721  20.698   
 27.812  20.669    26.903  20.640    25.993  20.613    25.084  20.587   
 24.175  20.562    23.266  20.539    22.357  20.516   ] scurve
 70.000  25.000 moveto 
[  70.000  25.000    71.220  25.292    72.357  25.576    73.493  25.875   
 74.629  26.188   ] scurve 
 15.000  40.000 moveto 
[  15.000  40.000    13.537  39.051    12.074  38.121    10.611  37.212   
  9.148  36.328     7.685  35.470     6.222  34.639     4.759  33.838   
  3.296  33.068     1.833  32.327     0.370  31.619    -1.093  30.941   
] scurve 
 15.000  40.000 moveto 
[  15.000  40.000    16.463  40.965    17.926  41.941    19.389  42.925   
 20.852  43.910    22.315  44.892    23.778  45.864    25.241  46.822   
 26.704  47.758    28.167  48.665    29.630  49.537    31.093  50.367   
 32.556  51.148    33.833  51.784    34.897  52.279    35.816  52.678   
 36.565  52.983    37.246  53.244    37.845  53.460    38.390  53.645   
 38.869  53.798    39.331  53.939    39.715  54.049    40.098  54.153   
 40.440  54.241    40.747  54.316    41.054  54.387    41.321  54.447   
 41.567  54.498    41.812  54.547    42.044  54.592    42.241  54.627   
 42.437  54.661    42.633  54.694    42.805  54.721    42.962  54.744   
 43.119  54.767    43.276  54.789    43.413  54.807    43.539  54.822   
 43.664  54.837    43.790  54.852    43.904  54.864    44.005  54.875   
 44.105  54.885    44.206  54.895    44.306  54.904    44.389  54.911   
 44.469  54.918    44.550  54.925    44.630  54.931    44.707  54.937   
 44.772  54.942    44.836  54.946    44.900  54.950    44.965  54.954   
 45.026  54.958    45.077  54.961    45.129  54.964    45.180  54.967   
 45.232  54.969    45.281  54.972    45.322  54.974    45.364  54.976   
] scurve 
  8.000  45.000 moveto 
[   8.000  45.000     6.545  44.024     5.090  43.044     3.636  42.066   
  2.181  41.095     0.726  40.134    -0.729  39.188   ] scurve 
  8.000  45.000 moveto 
[   8.000  45.000     9.455  45.966    10.910  46.916    12.364  47.844   
 13.819  48.744    15.274  49.608    16.729  50.430    18.184  51.202   
 19.453  51.831    20.490  52.310    21.387  52.698    22.132  53.000   
 22.809  53.259    23.405  53.473    23.947  53.656    24.424  53.808   
 24.883  53.947    25.265  54.056    25.646  54.159    25.986  54.247   
 26.291  54.321    26.596  54.392    26.868  54.452    27.112  54.503   
 27.356  54.551    27.587  54.595    27.782  54.630    27.977  54.664   
 28.172  54.696    28.343  54.723    28.499  54.746    28.655  54.769   
 28.811  54.790    28.948  54.808    29.073  54.824    29.198  54.839   
 29.323  54.853    29.439  54.866    29.539  54.876    29.639  54.886   
 29.738  54.896    29.838  54.905    29.920  54.912    30.000  54.919   
 30.080  54.925    30.160  54.932    30.237  54.937    30.301  54.942   
 30.365  54.946    30.429  54.951    30.493  54.955    30.554  54.958   
 30.605  54.961    30.656  54.964    30.707  54.967    30.758  54.970   
 30.808  54.972    30.849  54.974    30.890  54.976   ] scurve 
  0.000  20.000 moveto 
 20.000  70.000 moveto 
[  20.000  70.000    19.442  70.661    19.173  70.997    18.904  71.347   
 18.635  71.710    18.366  72.088    18.107  72.469    17.892  72.797   
 17.677  73.137    17.462  73.492    17.247  73.861    17.063  74.189   
 16.891  74.509    16.719  74.841    16.550  75.181    16.412  75.470   
 16.274  75.769    16.137  76.081    16.021  76.352    15.911  76.622   
 15.801  76.904    15.713  77.138    15.625  77.381    15.543  77.616   
 15.473  77.827    15.404  78.042    15.347  78.225    15.291  78.415   
 15.246  78.573    15.201  78.738    15.164  78.875    15.130  79.013   
 15.101  79.132    15.075  79.244    15.051  79.347    15.033  79.435   
 15.016  79.516    15.001  79.590    14.989  79.652    14.979  79.707   
 14.970  79.756    14.963  79.798    14.957  79.834    14.952  79.865
 14.949  79.891    14.945  79.913    14.943  79.931    14.941  79.946   
 14.939  79.958    14.938  79.967    14.937  79.975    14.937  79.982   
 14.936  79.986    14.936  79.990    14.935  79.993    14.935  79.995   
 14.935  79.996    14.935  79.997    14.935  79.998    14.935  79.999   
 14.935  79.999    14.935  79.999    14.935  80.000    14.935  80.000   
 14.935  80.000    14.935  80.000    14.935  80.000    14.935  80.000   
 14.935  80.000   ] scurve 
 20.000  70.000 moveto 
[  20.000  70.000    20.583  69.354    20.919  69.002    21.255  68.663   
 21.591  68.336    21.927  68.021    22.263  67.717    22.599  67.423   
 22.936  67.139    23.272  66.864    23.608  66.597    23.944  66.339   
 24.280  66.089    24.616  65.846    24.952  65.610    25.288  65.381   
 25.624  65.158    25.961  64.942    26.297  64.732    26.633  64.527   
 26.969  64.328    27.305  64.134    27.641  63.945    27.977  63.761   
 28.313  63.582    28.649  63.408    28.986  63.237    29.322  63.071   
 29.658  62.909    29.994  62.751    30.330  62.597    30.666  62.446   
 31.002  62.299    31.338  62.156    31.674  62.016    32.011  61.879   
 32.347  61.745    32.683  61.614    33.019  61.486    33.355  61.361   
 33.691  61.239    34.027  61.120    34.363  61.003    34.699  60.888   
 35.036  60.777    35.372  60.667    35.708  60.560    36.044  60.456   
 36.380  60.353    36.716  60.253    37.052  60.154    37.388  60.058   
 37.724  59.964    38.061  59.872    38.397  59.782    38.733  59.693   
 39.069  59.607    39.405  59.522    39.741  59.439    40.077  59.357   
 40.413  59.277    40.749  59.199    41.086  59.123    41.422  59.047   
 41.758  58.974    42.094  58.902    42.430  58.831    42.766  58.762   
 43.102  58.694    43.438  58.627    43.774  58.562    44.111  58.498   
 44.447  58.435    44.783  58.373    45.119  58.313    45.455  58.253   
 45.791  58.195    46.127  58.138    46.463  58.082    46.799  58.027   
 47.136  57.973    47.472  57.921    47.808  57.869    48.144  57.818   
 48.480  57.768    48.816  57.719    49.152  57.671    49.488  57.624   
 49.824  57.578    50.161  57.532    50.497  57.488    50.833  57.444   
 51.169  57.401    51.505  57.359    51.841  57.317    52.177  57.277   
 52.513  57.237    52.849  57.198    53.186  57.159    53.522  57.122   
 53.858  57.084    54.194  57.048    54.530  57.012    54.866  56.977   
 55.202  56.943    55.538  56.909    55.874  56.876    56.211  56.843   
 56.547  56.811    56.883  56.780    57.219  56.749    57.555  56.719   
 57.891  56.689    58.227  56.660    58.563  56.631    58.899  56.603   
 59.236  56.575    59.572  56.548    59.908  56.521    60.244  56.495   
 60.580  56.469    60.916  56.444    61.252  56.419    61.588  56.394   
 61.924  56.370    62.261  56.347    62.597  56.324    62.933  56.301   
 63.269  56.278    63.605  56.256    63.941  56.235    64.277  56.214   
 64.613  56.193    64.949  56.172    65.286  56.152    65.622  56.132   
 65.958  56.113    66.294  56.094    66.630  56.075    66.966  56.057   
 67.302  56.039    67.638  56.021    67.974  56.003    68.311  55.986   
 68.647  55.969    68.983  55.953    69.319  55.936    69.655  55.920   
 69.991  55.905    70.327  55.889    70.663  55.874    70.999  55.859   
 71.336  55.844    71.672  55.830    72.008  55.816    72.344  55.802   
 72.680  55.788    73.016  55.775    73.352  55.761    73.688  55.748   
 74.024  55.736    74.360  55.723   ] scurve 
  0.000  75.000 moveto 
[   0.000  75.000     0.410  74.211     0.608  73.857     0.805  73.517   
  1.002  73.190     1.200  72.874     1.397  72.570     1.595  72.275   
  1.792  71.989     1.990  71.712     2.187  71.443     2.385  71.182   
  2.582  70.928     2.780  70.681     2.977  70.441     3.175  70.206   
  3.372  69.978     3.569  69.755     3.767  69.537     3.964  69.324   
  4.162  69.116     4.359  68.913     4.557  68.715     4.754  68.520   
  4.952  68.330     5.149  68.144     5.347  67.961     5.544  67.783   
  5.742  67.608     5.939  67.436     6.136  67.268     6.334  67.103   
  6.531  66.941     6.729  66.782     6.926  66.626     7.124  66.473   
  7.321  66.322     7.519  66.175     7.716  66.030     7.914  65.887   
  8.111  65.747     8.309  65.610     8.506  65.474     8.704  65.341   
  8.901  65.211     9.098  65.082     9.296  64.956     9.493  64.831   
  9.691  64.709     9.888  64.588    10.086  64.470    10.283  64.353   
 10.481  64.238    10.678  64.125    10.876  64.014    11.073  63.904   
 11.271  63.796    11.468  63.690    11.665  63.585    11.863  63.482   
 12.060  63.381    12.258  63.280    12.455  63.182    12.653  63.085   
 12.850  62.989    13.048  62.894    13.245  62.801    13.443  62.710   
 13.640  62.619    13.838  62.530    14.035  62.442    14.232  62.355   
 14.430  62.270    14.627  62.186    14.825  62.102    15.022  62.020   
 15.220  61.940    15.417  61.860    15.615  61.781    15.812  61.703   
 16.010  61.627    16.207  61.551    16.405  61.477    16.602  61.403   
 16.800  61.330    16.997  61.259    17.194  61.188    17.392  61.118   
 17.589  61.049    17.787  60.981    17.984  60.914    18.182  60.848   
 18.379  60.783    18.577  60.718    18.774  60.654    18.972  60.591   
 19.169  60.529    19.367  60.468    19.564  60.407    19.761  60.347   
 19.959  60.288    20.156  60.230    20.354  60.172    20.551  60.115   
 20.749  60.059    20.946  60.004    21.144  59.949    21.341  59.895   
 21.539  59.841    21.736  59.788    21.934  59.736    22.131  59.684   
 22.328  59.634    22.526  59.583    22.723  59.533    22.921  59.484   
 23.118  59.436    23.316  59.388    23.513  59.340    23.711  59.294   
 23.908  59.247    24.106  59.201    24.303  59.156    24.501  59.112   
 24.698  59.067    24.896  59.024    25.093  58.981    25.290  58.938   
 25.488  58.896    25.685  58.854    25.883  58.813    26.080  58.773   
 26.278  58.732    26.475  58.693    26.673  58.653    26.870  58.615   
 27.068  58.576    27.265  58.538    27.463  58.501    27.660  58.464   
 27.857  58.427    28.055  58.391    28.252  58.355    28.450  58.320   
 28.647  58.285    28.845  58.250    29.042  58.216    29.240  58.182   
 29.437  58.148    29.635  58.115    29.832  58.083    30.030  58.050   
 30.227  58.018    30.424  57.987    30.622  57.955    30.819  57.924   
 31.017  57.894    31.214  57.864    31.412  57.834    31.609  57.804   
 31.807  57.775    32.004  57.746    32.202  57.717    32.399  57.689   
 32.597  57.661    32.794  57.633    32.992  57.606    33.189  57.579   
 33.386  57.552    33.593  57.525    33.840  57.492    34.087  57.460   
 34.333  57.428    34.580  57.396    34.827  57.365    35.074  57.335   
 35.321  57.305    35.568  57.275    35.814  57.246    36.061  57.217   
 36.308  57.188    36.555  57.160    36.802  57.132    37.048  57.105   
 37.295  57.078    37.542  57.051    37.789  57.025    38.036  56.999   
 38.283  56.973    38.529  56.948   ] scurve
[  38.529  56.948    38.776  56.923    39.023  56.899    39.270  56.874   
 39.517  56.850    39.764  56.827    40.010  56.804    40.257  56.780   
 40.504  56.758    40.751  56.735    40.998  56.713    41.245  56.691   
 41.491  56.670    41.738  56.649    41.985  56.628    42.232  56.607   
 42.479  56.587    42.726  56.566    42.972  56.546    43.219  56.527   
 43.466  56.507    43.713  56.488    43.960  56.469    44.207  56.451   
 44.453  56.432    44.700  56.414    44.947  56.396    45.194  56.379   
 45.441  56.361    45.688  56.344    45.934  56.327    46.181  56.310   
 46.428  56.294    46.675  56.277    46.922  56.261    47.168  56.245   
 47.415  56.229    47.662  56.214    47.909  56.199    48.156  56.184   
 48.403  56.169    48.649  56.154    48.896  56.139    49.143  56.125   
 49.390  56.111    49.637  56.097    49.884  56.083    50.130  56.069   
 50.377  56.056    50.624  56.043    50.871  56.029    51.118  56.016   
 51.365  56.004    51.611  55.991    51.858  55.979    52.105  55.966   
 52.352  55.954    52.599  55.942    52.846  55.930    53.092  55.919   
 53.339  55.907    53.586  55.896    53.833  55.884   ] scurve
 %%% . 
 36.401  75.000 moveto 
[  36.401  75.000    36.008  75.831    35.881  76.119    35.755  76.420   
 35.651  76.675    35.550  76.935    35.453  77.196    35.372  77.421   
 35.291  77.655    35.221  77.868    35.156  78.071    35.097  78.263   
 35.045  78.438    34.997  78.608    34.956  78.760    34.918  78.904   
 34.885  79.036    34.857  79.154    34.831  79.266    34.810  79.361   
 34.791  79.450    34.774  79.531    34.760  79.599    34.749  79.662   
 34.738  79.717    34.730  79.764    34.723  79.806    34.718  79.840   
 34.713  79.870    34.709  79.895    34.706  79.916    34.704  79.934   
 34.702  79.948    34.700  79.960    34.699  79.969    34.698  79.976   
 34.698  79.982    34.697  79.987    34.697  79.990    34.697  79.993   
 34.696  79.995    34.696  79.997    34.696  79.998    34.696  79.998   
 34.696  79.999    34.696  79.999    34.696  80.000    34.696  80.000   
 34.696  80.000    34.696  80.000    34.696  80.000    34.696  80.000   
 34.696  80.000   ] scurve 
 36.401  75.000 moveto 
[  36.401  75.000    36.811  74.211    37.009  73.857    37.206  73.517   
 37.403  73.190    37.601  72.874    37.798  72.570    37.996  72.275   
 38.193  71.989    38.391  71.712    38.588  71.443    38.786  71.182   
 38.983  70.928    39.181  70.681    39.378  70.441    39.576  70.206   
 39.773  69.978    39.971  69.755    40.168  69.537    40.365  69.324   
 40.563  69.116    40.760  68.913    40.958  68.715    41.155  68.520   
 41.353  68.330    41.550  68.144    41.748  67.961    41.945  67.783   
 42.143  67.608    42.340  67.436    42.538  67.268    42.735  67.103   
 42.932  66.941    43.130  66.782    43.327  66.626    43.525  66.473   
 43.722  66.322    43.920  66.175    44.117  66.030    44.315  65.887   
 44.512  65.747    44.710  65.610    44.907  65.474    45.105  65.341   
 45.302  65.211    45.499  65.082    45.697  64.956    45.894  64.831   
 46.092  64.709    46.289  64.588    46.487  64.470    46.684  64.353   
 46.882  64.238    47.079  64.125    47.277  64.014    47.474  63.904   
 47.672  63.796    47.869  63.690    48.067  63.585    48.264  63.482   
 48.461  63.381    48.659  63.280    48.856  63.182    49.054  63.085   
 49.251  62.989    49.449  62.894    49.646  62.801    49.844  62.710   
 50.041  62.619    50.239  62.530    50.436  62.442    50.634  62.355   
 50.831  62.270    51.028  62.186    51.226  62.102    51.423  62.020   
 51.621  61.940    51.818  61.860    52.016  61.781    52.213  61.703   
 52.411  61.627    52.608  61.551    52.806  61.477    53.003  61.403   
 53.201  61.330    53.398  61.259    53.595  61.188    53.793  61.118   
 53.990  61.049    54.188  60.981    54.385  60.914    54.583  60.848   
 54.780  60.783    54.978  60.718    55.175  60.654    55.373  60.591   
 55.570  60.529    55.768  60.468    55.965  60.407    56.163  60.347   
 56.360  60.288    56.557  60.230    56.755  60.172    56.952  60.115   
 57.150  60.059    57.347  60.004    57.545  59.949    57.742  59.895   
 57.940  59.841    58.137  59.788    58.335  59.736    58.532  59.684   
 58.730  59.634    58.927  59.583    59.124  59.533    59.322  59.484   
 59.519  59.436    59.717  59.388    59.914  59.340    60.112  59.294   
 60.309  59.247    60.507  59.201    60.704  59.156    60.902  59.112   
 61.099  59.067    61.297  59.024    61.494  58.981    61.691  58.938   
 61.889  58.896    62.086  58.854    62.284  58.813    62.481  58.773   
 62.679  58.732    62.876  58.693    63.074  58.653    63.271  58.615   
 63.469  58.576    63.666  58.538    63.864  58.501    64.061  58.464   
 64.259  58.427    64.456  58.391    64.653  58.355    64.851  58.320   
 65.048  58.285    65.246  58.250    65.443  58.216    65.641  58.182   
 65.838  58.148    66.036  58.115    66.233  58.083    66.431  58.050   
 66.628  58.018    66.826  57.987    67.023  57.955    67.220  57.924   
 67.418  57.894    67.615  57.864    67.813  57.834    68.010  57.804   
 68.208  57.775    68.405  57.746    68.603  57.717    68.800  57.689   
 68.998  57.661    69.195  57.633    69.393  57.606    69.590  57.579   
 69.787  57.552    69.994  57.525    70.241  57.492    70.488  57.460   
 70.734  57.428    70.981  57.396    71.228  57.365    71.475  57.335   
 71.722  57.305    71.969  57.275    72.215  57.246    72.462  57.217   
 72.709  57.188    72.956  57.160    73.203  57.132    73.450  57.105   
 73.696  57.078    73.943  57.051    74.190  57.025   ] scurve 
 58.554  70.353 moveto 
[  58.554  70.353    58.006  71.027    57.742  71.371    57.478  71.729   
 57.214  72.101    56.950  72.488    56.729  72.826    56.518  73.161   
 56.307  73.510    56.096  73.873    55.904  74.217    55.735  74.532   
 55.566  74.859    55.397  75.199    55.256  75.496    55.121  75.791   
 54.986  76.098    54.868  76.377    54.760  76.643    54.651  76.919   
 54.561  77.160    54.475  77.400    54.393  77.636    54.324  77.844   
 54.254  78.060    54.199  78.240    54.144  78.427    54.098  78.586   
 54.054  78.749    54.018  78.887    53.983  79.025    53.955  79.142   
 53.929  79.252    53.906  79.354    53.888  79.440    53.871  79.521   
 53.856  79.594    53.845  79.655    53.835  79.711    53.826  79.759   
 53.819  79.801    53.813  79.837    53.809  79.867    53.805  79.893   
 53.802  79.914    53.799  79.932    53.797  79.947    53.796  79.958   
 53.795  79.968    53.794  79.976    53.793  79.982    53.792  79.986   
 53.792  79.990    53.792  79.993    53.792  79.995    53.791  79.996   
 53.791  79.997    53.791  79.998    53.791  79.999    53.791  79.999   
 53.791  80.000    53.791  80.000    53.791  80.000    53.791  80.000   
 53.791  80.000    53.791  80.000    53.791  80.000    53.791  80.000   
] scurve 
 58.554  70.353 moveto 
[  58.554  70.353    59.126  69.694    59.456  69.335    59.786  68.990   
 60.116  68.657    60.446  68.337    60.776  68.028    61.106  67.729   
 61.436  67.440    61.766  67.160    62.095  66.889    62.425  66.627   
 62.755  66.372    63.085  66.126    63.415  65.886    63.745  65.653   
 64.075  65.427    64.405  65.208    64.735  64.994    65.065  64.786   
 65.395  64.584    65.725  64.387    66.055  64.195    66.385  64.008   
 66.715  63.826    67.044  63.648    67.374  63.475    67.704  63.306   
 68.034  63.142    68.364  62.981    68.694  62.824    69.024  62.671   
 69.354  62.521    69.684  62.375    70.014  62.232    70.344  62.093   
 70.674  61.957    71.004  61.824    71.334  61.693    71.663  61.566   
 71.993  61.442    72.323  61.320    72.653  61.201    72.983  61.085   
 73.313  60.971    73.643  60.859    73.973  60.750    74.303  60.643   
] scurve 
grestore}
\LabelTeX   77.500  31.300 $x$\ELTX
\LabelTeX   26.500  78.000 $y$\ELTX
\LabelTeX   86.000  54.000 $y=y_2$\ELTX
\LabelTeX   86.000  19.000 $y=y_1$\ELTX
\LabelTeX   86.000  66.500 $g(y)<0$\ELTX
\LabelTeX   86.000  37.000 $g(y)>0$\ELTX
\LabelTeX   86.000  10.000 $g(y)<0$\ELTX
\EndFig
\vskip8mm

\noindent {\tenpointbf c) Cas g\'en\'eral des \'equations \`a variables 
s\'epar\'ees~:}
$$
\hbox{$y'=f(x)g(y)$ avec $f,\,g$ continues.}
\leqno{\rm(E)}
$$
\smallskip
\noindent$\bullet$ Si $g(y_j)=0$, la fonction constante $y(x)=y_j$ est
solution singuli\`ere. 
\vskip3pt
\noindent$\bullet$ Sur l'ouvert $U=\{(x,y)\,;\,\,g(y)\not=0\}$ on a
$$
{\rm (E)}\Leftrightarrow\frac{dy}{g(y)}=f(x)dx
$$
d'o\`u $G(y)=F(x)+\lambda$,\quad$\lambda\in\bR$, o\`u $F$ est une primitive 
de $f$ et $G$ une primitive de~$1/g$.
Comme $G$ est continue strictement monotone sur chaque intervalle
$[y_j,y_{j+1}[$, l'application $G$ admet une application r\'eciproque
$G^{-1}$ et on obtient
$$
y=G^{-1}(F(x)+\lambda).
$$

\bigskip
\state Exemple -- |  Soit l'\'equation $\displaystyle 
y'=\sqrt{\dfrac{1-y^2}{1-x^2}}$. Le domaine de d\'efinition est la r\'eunion
$$
\{|x|<1\ {\rm et}\  |y|\leq 1\}\quad\cup\quad\{|x|>1\ {\rm et}\ |y|\geq 1\}.
$$
On va donc se placer dans l'ouvert 
$$
U=\{|x|<1\ {\rm et}\  |y|<1\}\quad\cup\quad\{|x|>1\ {\rm et}\  |y|>1\}.
$$

\noindent$\bullet$ Dans le carr\'e $\{|x|<1$ et $|y|<1\}$ l'\'equation 
s'\'ecrit~:
$$
\frac{dy}{\sqrt{1-y^2}}=\frac{dx}{\sqrt{1-x^2}},
$$
d'o\`u $\Arcsin y=\Arcsin x+\lambda$,\quad$\lambda\in\bR$.
Comme Arcsin est une bijection de
$]-1,1[$ sur $\big]-\dfrac{\pi}{2},\dfrac{\pi}{2}\big[$, on a n\'ecessairement
$\lambda\in{}]-\pi,\pi[$. On doit avoir de plus 
$$
\Arcsin x\in\Big]-\frac{\pi}{2},\frac{\pi}{2}\Big[\cap\Big]-\frac{\pi}{2}
-\lambda,\frac{\pi}{2}-\lambda\Big[{}={}\left\{
\eqalign{\Big]-\dfrac{\pi}{2},\dfrac{\pi}{2}-\lambda
\Big[\quad{\rm si}\quad\lambda\geq 0,\hfill\cr 
\noalign{\vskip3pt}
\Big]-\dfrac{\pi}{2}-\lambda,\dfrac{\pi}{2}\Big[
\quad{\rm si}\quad\lambda\leq 0.\hfill\cr}
\right.
$$
De m\^eme $\Arcsin y$ est dans $\big]-\dfrac{\pi}{2}+\lambda,\dfrac{\pi}{2}
\big[$ si $\lambda\geq 0$, et dans $\big]-\dfrac{\pi}{2},\dfrac{\pi}{2}
+\lambda\big[$ si $\lambda\leq 0$.

\item{} Les courbes int\'egrales admettent pour \'equation
$$
y=\sin(\Arcsin x+\lambda)=x\,\cos\,\lambda +\sqrt{1-x^2}\sin\,\lambda
$$
avec
$$
\eqalign{
x&\in{}]-1,\cos\,\lambda[,\quad y\in{}]-\cos\,\lambda,1[\quad{\rm si}\quad 
\lambda\geq 0,\cr
x&\in{}]-\cos\,\lambda,1[,\quad y\in{}]-1,\cos\,\lambda[\quad{\rm si}\quad 
\lambda\leq 0.\cr}
$$
L'\'equation ci-dessus implique $(y-x\,\cos\,\lambda )^2+x^2\,\sin^2\lambda
=\sin^2\lambda$, donc les courbes int\'egrales sont des arcs d'ellipse.

\medskip
\noindent$\bullet$ L'ouvert $\{|x|>1$ et $|y|>1\}$ est form\'e de 4
composantes connexes. Pla\c{c}ons-nous par exemple dans $\{x>1$ et 
$y>1\}$. On a
$$
(E)\Leftrightarrow\frac{dy}{\sqrt{y^2-1}}=\frac{dx}{\sqrt{x^2-1}},
$$
d'o\`u $\Argcosh y=\Argcosh x+\lambda$, $\lambda\in\bR$. $\Argcosh$ est une
bijection de $]1,+\infty[$ sur $]0,+\infty[$~; en raisonnant comme ci-dessus,
on obtient
$$
y=x\,\cosh\,\lambda +\sqrt{x^2-1}\,\sinh\,\lambda
$$
avec
$$
\eqalign{
x&\in{}]1,+\infty[,\quad y\in{}]\cosh\,\lambda,+\infty[\quad{\rm si}\quad
\lambda\geq 0,\cr 
x&\in{}]\cosh\,\lambda,+\infty[,\quad y\in{}]1,+\infty[\quad{\rm si}\quad
\lambda\leq 0,\cr
}
$$
par suite $(y-x\,\cosh\,\lambda )^2-x^2\,\sinh^2\lambda+\sinh^2\lambda=0$,
ce qui est l'\'equation d'une conique. Comme
$\sqrt{x^2-1}=|x|\,\sqrt{1-\dfrac{1}{x^2}}=
|x|-\dfrac{1}{2|x|}+O\big(\dfrac{1}{x^3}\big)$,
on voit que la conique admet des asymptotes $y=(\cosh\,\lambda\pm
\sinh\,\lambda)x=e^{\pm\lambda}x$ (pour la branche $x>1$ qui nous
int\'eresse, c'est $y=e^{\lambda}x$). On a donc affaire \`a des
arcs d'hyperbole.

\InsertFig 0.000 107.000 {
1 mm unit /nbpt 10 def
gsave 
 10.000  10.000 moveto 
110.000  90.000 rect  clip  newpath
 10.000  40.000 moveto 
 40.000  30.000 rect 
closepath 
  0.940 setgray  fill 0 setgray 
 50.000  40.000 moveto 
 30.000 -30.000 rect 
closepath 
  0.940 setgray  fill 0 setgray 
 80.000  40.000 moveto 
 40.000  30.000 rect 
closepath 
  0.940 setgray  fill 0 setgray 
 80.000  70.000 moveto 
-30.000  30.000 rect 
closepath 
  0.940 setgray  fill 0 setgray 
 10.000  55.000 moveto 
110.000   0.000 vector 
 65.000  10.000 moveto 
 90.000  90.000 vector 
 10.000  40.000 moveto 
120.000  40.000 lineto 
stroke 
120.000  70.000 moveto 
 10.000  70.000 lineto 
stroke 
 50.000 100.000 moveto 
 50.000  10.000 lineto 
stroke 
 80.000  10.000 moveto 
 80.000 100.000 lineto 
stroke
  0.340 setlinewidth
 68.892  55.000 moveto
  0.500  90.000 indent
 72.608  55.000 moveto
  0.500  90.000 indent
 84.332  55.000 moveto
  0.500  90.000 indent
 65.000  77.625 moveto
  0.500   0.000 indent
 65.000  51.031 moveto
  0.500   0.000 indent
 65.000  47.450 moveto
  0.500   0.000 indent  
 56.000  64.000 moveto 
[  56.000  64.000    54.681  62.562    53.498  61.023    52.468  59.405   
 51.604  57.728    50.917  56.015    50.417  54.289    50.110  52.571   
 50.002  51.031  ]  scurve
 56.000  64.000 moveto 
[  56.000  64.000    57.439  65.319    58.977  66.501    60.566  67.513   
 61.731  68.136    62.552  68.518    63.202  68.789    63.734  68.991   
 64.186  69.147    64.564  69.268    64.913  69.372    65.203  69.452   
 65.477  69.523    65.709  69.579    65.940  69.631    66.125  69.671   
 66.310  69.708    66.482  69.741    66.629  69.768    66.777  69.793   
 66.908  69.814    67.025  69.832    67.143  69.849    67.252  69.865   
 67.346  69.877    67.440  69.889    67.533  69.900    67.611  69.909   
 67.686  69.917    67.761  69.925    67.834  69.932    67.894  69.938   
 67.954  69.943    68.013  69.948    68.073  69.953    68.120  69.957   
 68.168  69.960    68.215  69.964    68.263  69.967    68.304  69.970   
 68.342  69.972    68.380  69.974    68.418  69.977    68.454  69.979   
 68.485  69.980    68.515  69.982    68.545  69.983    68.575  69.985   
 68.603  69.986    68.627  69.987    68.651  69.988    68.675  69.989   
 68.699  69.990    68.721  69.991    68.741  69.991    68.760  69.992   
 68.779  69.993    68.799  69.993    68.816  69.994    68.831  69.994   
 68.847  69.995    68.862  69.995    68.878  69.995    68.892  69.996   
] scurve 
 78.000  42.000 moveto 
[  78.000  42.000    76.853  41.099    76.360  40.825    76.008  40.662   
 75.722  40.546    75.483  40.459    75.272  40.391    75.098  40.339   
 74.937  40.295    74.793  40.258    74.659  40.227    74.542  40.201   
 74.424  40.177    74.329  40.159    74.233  40.142    74.142  40.127   
 74.065  40.114    73.987  40.103    73.911  40.092    73.849  40.084   
 73.785  40.076    73.722  40.068    73.669  40.062    73.618  40.057   
 73.567  40.051    73.516  40.046    73.475  40.043    73.433  40.039   
 73.392  40.035    73.351  40.032    73.317  40.030    73.284  40.027   
 73.251  40.025    73.217  40.023    73.188  40.021    73.161  40.019   
 73.135  40.018    73.108  40.016    73.081  40.015    73.060  40.014   
 73.038  40.012    73.017  40.011    72.995  40.011    72.975  40.010   
 72.958  40.009    72.940  40.008    72.923  40.008    72.906  40.007   
 72.890  40.006    72.876  40.006    72.862  40.006    72.848  40.005   
 72.834  40.005    72.821  40.004    72.810  40.004    72.798  40.004   
 72.787  40.003    72.776  40.003    72.765  40.003    72.756  40.003   
 72.747  40.003    72.738  40.002    72.729  40.002    72.720  40.002   
 72.712  40.002    72.705  40.002    72.698  40.002    72.691  40.002   
 72.683  40.001    72.676  40.001    72.671  40.001    72.665  40.001   
 72.659  40.001    72.653  40.001    72.648  40.001    72.643  40.001   
 72.638  40.001    72.633  40.001    72.629  40.001    72.624  40.001   
 72.619  40.001    72.616  40.001    72.612  40.001    72.608  40.000   
] scurve 
 78.000  42.000 moveto 
[  78.000  42.000    79.025  43.355    79.460  44.286    79.720  45.109   
 79.881  45.891    79.968  46.628    79.999  47.310    80.000  47.450
] scurve
 35.000  87.000 moveto 
[  35.000  87.000    33.637  85.601    32.192  84.281    30.662  83.037   
 29.043  81.867    27.330  80.766    25.520  79.732    23.607  78.763   
 21.587  77.855    19.453  77.007    17.202  76.217    14.826  75.481   
 12.554  74.858    10.461  74.346     8.319  73.874   ] scurve 
 35.000  87.000 moveto 
[  35.000  87.000    36.285  88.482    37.496  90.051    38.636  91.711   
 39.707  93.466    40.712  95.321    41.654  97.281    42.536  99.350   
 43.359 101.533   ] scurve 
 45.000  75.000 moveto 
[  45.000  75.000    43.506  73.725    41.971  72.759    40.889  72.225   
 40.001  71.857    39.253  71.587    38.575  71.371    37.969  71.198   
 37.413  71.054    36.899  70.933    36.410  70.828    35.979  70.742   
 35.539  70.661    35.148  70.594    34.783  70.536    34.412  70.481   
 34.084  70.436    33.778  70.396    33.469  70.358    33.162  70.323   
 32.909  70.296    32.653  70.270    32.394  70.245    32.147  70.222   
 31.937  70.204    31.724  70.187    31.510  70.170    31.298  70.155   
 31.124  70.143    30.949  70.131    30.773  70.120    30.596  70.110   
 30.440  70.101    30.297  70.093    30.153  70.086    30.008  70.079   
 29.862  70.072    29.745  70.067    29.628  70.061    29.510  70.057   
 29.391  70.052   ] scurve 
 45.000  75.000 moveto 
[  45.000  75.000    46.275  76.493    47.346  78.220    48.066  79.791   
 48.554  81.165    48.913  82.433    49.201  83.700    49.432  84.974   
 49.596  86.112    49.732  87.306    49.840  88.560    49.920  89.875   
 49.967  91.074    49.992  92.218    50.000  93.362  ]  scurve
 39.000  81.000 moveto 
[  39.000  81.000    37.562  79.672    36.008  78.449    34.330  77.326   
 32.523  76.297    30.578  75.359    28.606  74.552    26.829  73.927   
 25.299  73.453    23.762  73.030    22.459  72.708    21.109  72.405   
 19.917  72.161    18.767  71.946    17.585  71.743    16.548  71.578   
 15.554  71.432    14.538  71.293    13.576  71.171    12.729  71.071   
 11.867  70.976    10.990  70.885    10.195  70.808     9.470  70.741   
] scurve 
 39.000  81.000 moveto 
[  39.000  81.000    40.328  82.438    41.551  83.992    42.675  85.668   
 43.705  87.474    44.644  89.418    45.497  91.507    46.268  93.750   
 46.959  96.157    47.563  98.689    48.002 100.884   ] scurve 
 25.000  91.000 moveto 
[  25.000  91.000    23.500  89.732    21.935  88.519    20.302  87.358   
 18.600  86.249    16.824  85.188    14.973  84.174    13.044  83.207   
 11.033  82.284     8.937  81.403   ] scurve 
 25.000  91.000 moveto 
[  25.000  91.000    26.437  92.324    27.814  93.707    29.133  95.151   
 30.395  96.657    31.603  98.229    32.759  99.868    33.864 101.578   
] scurve 
 17.000  95.000 moveto 
[  17.000  95.000    15.448  93.783    13.842  92.609    12.177  91.477   
 10.454  90.386     8.670  89.334   ] scurve 
 17.000  95.000 moveto 
[  17.000  95.000    18.498  96.262    19.944  97.571    21.339  98.927   
 22.686 100.332   ] scurve 
 85.000  35.000 moveto 
[  85.000  35.000    83.725  33.507    82.654  31.780    81.934  30.209   
 81.446  28.835    81.087  27.567    80.799  26.300    80.568  25.026   
 80.404  23.888    80.268  22.694    80.160  21.440    80.080  20.125   
 80.033  18.926    80.008  17.782    80.000  16.638   ] scurve
 85.000  35.000 moveto 
[  85.000  35.000    86.494  36.275    88.029  37.241    89.111  37.775   
 89.999  38.143    90.747  38.413    91.425  38.629    92.031  38.802   
 92.587  38.946    93.101  39.067    93.590  39.172    94.021  39.258   
 94.461  39.339    94.852  39.406    95.217  39.464    95.588  39.519   
 95.916  39.564    96.222  39.604    96.531  39.642    96.838  39.677   
 97.091  39.704    97.347  39.730    97.606  39.755    97.853  39.778   
 98.063  39.796    98.276  39.813    98.490  39.830    98.702  39.845   
 98.876  39.857    99.051  39.869    99.227  39.880    99.404  39.890   
 99.560  39.899    99.703  39.907    99.847  39.914    99.992  39.921   
100.138  39.928   100.255  39.933   100.372  39.939   100.490  39.943   
100.609  39.948   100.726  39.952   100.821  39.956   100.917  39.959   
101.013  39.962   101.109  39.965   101.206  39.968   101.284  39.970   
101.362  39.972   101.440  39.974   101.518  39.976   101.596  39.978   
101.664  39.980   101.727  39.981   101.790  39.983   101.853  39.984   
101.916  39.985   ] scurve 
 91.000  29.000 moveto 
[  91.000  29.000    89.672  27.562    88.449  26.008    87.325  24.332   
 86.295  22.526    85.356  20.582    84.503  18.493    83.732  16.250   
 83.041  13.843    82.437  11.311    81.998   9.116   ] scurve 
 91.000  29.000 moveto 
[  91.000  29.000    92.438  30.328    93.992  31.551    95.670  32.674   
 97.477  33.703    99.422  34.641   101.394  35.448   103.171  36.073   
104.701  36.547   106.238  36.970   107.541  37.292   108.891  37.595   
110.083  37.839   111.233  38.054   112.415  38.257   113.452  38.422   
114.446  38.568   115.462  38.707   116.424  38.829   117.271  38.929   
118.133  39.024   119.010  39.115   119.805  39.192   120.530  39.259   
] scurve 
 99.000  25.000 moveto 
[  99.000  25.000    97.547  23.692    96.172  22.310    94.872  20.849   
 93.643  19.307    92.483  17.679    91.389  15.962    90.358  14.152   
 89.387  12.244    88.475  10.234    87.620   8.117   ] scurve 
 99.000  25.000 moveto 
[  99.000  25.000   100.534  26.235   102.154  27.402   103.862  28.502   
105.663  29.539   107.562  30.515   109.562  31.432   111.670  32.292   
113.889  33.098   116.225  33.851   118.684  34.554   121.030  35.150   
] scurve 
107.000  21.000 moveto 
[ 107.000  21.000   105.478  19.763   104.019  18.472   102.622  17.123   
101.284  15.714   100.002  14.244    98.776  12.710    97.603  11.109   
 96.481   9.440   ] scurve 
107.000  21.000 moveto 
[ 107.000  21.000   108.588  22.183   110.245  23.314   111.973  24.395   
113.774  25.428   115.652  26.414   117.610  27.356   119.651  28.254   
121.777  29.110   ] scurve 
113.000  15.000 moveto 
[ 113.000  15.000   111.502  13.738   110.056  12.429   108.661  11.073   
107.314   9.668   ] scurve 
113.000  15.000 moveto 
[ 113.000  15.000   114.552  16.217   116.158  17.391   117.823  18.523   
119.546  19.614   121.330  20.666   ] scurve 
 94.000 100.000 moveto 
[  94.000 100.000    93.004  98.293    92.055  96.659    91.152  95.096   
 90.294  93.601    89.478  92.171    88.703  90.803    87.968  89.496   
 87.273  88.248    86.615  87.056    85.993  85.918    85.407  84.832   
 84.855  83.796    84.337  82.810    83.852  81.870    83.398  80.976   
 82.976  80.125    82.584  79.318    82.221  78.551    81.888  77.824   
 81.583  77.136    81.306  76.485    81.057  75.870    80.835  75.291   
 80.640  74.746    80.471  74.234    80.328  73.755    80.213  73.316   
 80.138  72.980    80.082  72.680    80.046  72.440    80.022  72.228   
 80.008  72.053    80.001  71.904    80.000  71.788   ] scurve
 94.000 100.000 moveto 
[  94.000 100.000    95.045 101.782    96.141 103.643   ] scurve 
120.000  83.000 moveto 
[ 120.000  83.000   118.125  82.165   116.319  81.366   114.580  80.600   
112.906  79.869   111.294  79.169   109.743  78.501   108.250  77.864   
106.813  77.256   105.431  76.677   104.101  76.127   102.823  75.604   
101.594  75.108   100.412  74.638    99.277  74.194    98.186  73.774   
 97.139  73.379    96.133  73.009    95.168  72.661    94.243  72.336   
 93.355  72.035    92.505  71.755    91.763  71.519    91.134  71.327   
 90.526  71.148    90.035  71.008    89.578  70.884    89.151  70.772   
 88.805  70.685    88.467  70.603    88.161  70.532    87.903  70.474   
 87.649  70.420    87.419  70.372    87.223  70.334    87.031  70.297   
 86.852  70.265    86.702  70.239    86.555  70.214    86.409  70.191   
 86.294  70.173    86.180  70.156    86.068  70.140    85.966  70.127   
 85.878  70.115    85.791  70.104    85.704  70.094    85.630  70.085   
 85.562  70.078    85.494  70.071    85.426  70.064    85.370  70.058   
 85.317  70.053    85.264  70.049    85.211  70.044    85.167  70.040   
 85.125  70.037    85.083  70.034    85.042  70.031    85.004  70.028   
 84.971  70.026    84.939  70.024    84.906  70.022    84.874  70.020   
 84.847  70.018    84.821  70.017    84.795  70.015    84.770  70.014   
 84.745  70.013    84.725  70.012    84.704  70.011    84.684  70.010   
 84.664  70.009    84.645  70.009    84.629  70.008    84.613  70.007   
 84.597  70.007    84.581  70.006    84.565  70.006    84.552  70.005   
 84.540  70.005    84.527  70.005    84.514  70.004    84.502  70.004   
 84.491  70.004    84.481  70.003    84.471  70.003    84.461  70.003   
 84.451  70.003    84.443  70.002    84.434  70.002    84.426  70.002   
 84.418  70.002    84.410  70.002    84.403  70.002    84.396  70.002   
 84.390  70.001    84.384  70.001    84.377  70.001    84.371  70.001   
 84.366  70.001    84.360  70.001    84.355  70.001    84.350  70.001   
 84.345  70.001    84.340  70.001    84.336  70.001    84.332  70.001   
] scurve 
 84.000  97.000 moveto 
[  84.000  97.000    83.394  94.929    82.843  92.976    82.344  91.135   
 81.897  89.400    81.499  87.768    81.150  86.232    80.848  84.788   
 80.615  83.553    80.441  82.523    80.296  81.545    80.181  80.618   
 80.106  79.880    80.053  79.204    80.018  78.558    80.003  78.039   
 80.000  77.625  ] scurve
 84.000  97.000 moveto 
[  84.000  97.000    84.662  99.192    85.381 101.515   ] scurve 
 36.000  10.000 moveto 
[  36.000  10.000    34.955   8.218    33.859   6.357   ] scurve 
 36.000  10.000 moveto 
[  36.000  10.000    36.996  11.707    37.945  13.341    38.848  14.904   
 39.706  16.399    40.522  17.829    41.297  19.197    42.032  20.504   
 42.727  21.752    43.385  22.944    44.007  24.082    44.593  25.168   
 45.145  26.204    45.663  27.190    46.148  28.130    46.602  29.024   
 47.024  29.875    47.416  30.682    47.779  31.449    48.112  32.176   
 48.417  32.864    48.694  33.515    48.943  34.130    49.165  34.709   
 49.360  35.254    49.529  35.766    49.672  36.245    49.787  36.684   
 49.862  37.020    49.918  37.320    49.954  37.560    49.978  37.772   
 49.992  37.947    49.999  38.096    50.000  38.212   ] scurve
 20.000  26.000 moveto 
[  20.000  26.000    18.218  24.954    16.356  23.858    14.412  22.709   
 12.383  21.505    10.265  20.245     8.055  18.926   ] scurve 
 20.000  26.000 moveto 
[  20.000  26.000    21.706  26.995    23.340  27.944    24.902  28.846   
 26.398  29.704    27.827  30.520    29.194  31.294    30.501  32.029   
 31.749  32.724    32.941  33.382    34.079  34.003    35.165  34.589   
 36.200  35.141    37.186  35.659    38.126  36.144    39.020  36.597   
 39.870  37.020    40.678  37.412    41.445  37.774    42.172  38.107   
 42.860  38.412    43.511  38.688    44.016  38.894    44.487  39.078   
 44.848  39.213    45.189  39.334    45.457  39.426    45.717  39.510   
 45.918  39.573    46.114  39.631    46.277  39.678    46.426  39.718   
 46.564  39.754    46.679  39.783    46.791  39.809    46.887  39.831   
 46.974  39.850    47.060  39.868    47.128  39.881    47.195  39.894   
 47.261  39.906    47.315  39.915    47.367  39.924    47.417  39.932   
 47.461  39.939    47.501  39.944    47.541  39.950    47.579  39.955   
 47.611  39.959    47.642  39.963    47.672  39.967    47.700  39.970   
 47.724  39.972    47.748  39.975    47.773  39.977    47.794  39.979   
 47.813  39.981    47.832  39.983    47.851  39.984    47.869  39.986   
 47.884  39.987    47.898  39.988    47.913  39.989    47.928  39.990   
 47.940  39.991    47.952  39.991   ] scurve 
 12.000  31.000 moveto 
[  12.000  31.000     9.938  30.235     7.789  29.432   ] scurve 
 12.000  31.000 moveto 
[  12.000  31.000    13.978  31.725    15.876  32.413    17.697  33.065   
 19.442  33.683    21.116  34.267    22.720  34.819    24.258  35.338   
 25.731  35.827    27.143  36.286    28.494  36.715    29.788  37.115   
 31.026  37.487    32.105  37.801    33.020  38.059    33.901  38.300   
 34.676  38.505    35.336  38.674    35.977  38.832    36.555  38.969   
 37.041  39.081    37.515  39.186    37.953  39.279    38.316  39.353   
 38.671  39.423    39.014  39.488    39.288  39.538    39.558  39.585   
 39.823  39.630    40.046  39.666    40.253  39.698    40.456  39.729   
 40.650  39.757    40.809  39.779    40.967  39.800    41.122  39.820   
 41.262  39.837    41.384  39.851    41.505  39.865    41.625  39.878   
 41.731  39.889    41.826  39.898    41.920  39.907    42.013  39.916   
 42.097  39.924    42.171  39.930    42.244  39.936    42.317  39.942   
 42.387  39.947    42.444  39.951    42.502  39.955    42.559  39.959   
 42.616  39.963    42.664  39.966    42.709  39.968    42.754  39.971   
 42.799  39.973    42.843  39.976    42.879  39.977    42.914  39.979   
 42.950  39.981    42.985  39.982    43.019  39.984    43.047  39.985   
 43.075  39.986    43.103  39.987    43.131  39.988   ] scurve 
 45.000  10.000 moveto 
[  45.000  10.000    44.318   7.838    43.585   5.559   ] scurve 
 45.000  10.000 moveto 
[  45.000  10.000    45.633  12.054    46.217  14.001    46.756  15.848   
 47.249  17.597    47.699  19.254    48.106  20.823    48.471  22.308   
 48.796  23.712    49.080  25.038    49.326  26.291    49.532  27.473   
 49.679  28.428    49.791  29.279    49.879  30.090    49.939  30.806   
 49.974  31.398    49.995  31.966    50.000  32.415   ] scurve
 65.000  55.000 moveto 
110.000 100.000 lineto 
stroke 
 65.000  55.000 moveto 
 20.000  10.000 lineto 
stroke 
  0.113 setlinewidth 
[   6.000   1.500   0.750   1.500 ] 0 setdash 
 65.000  55.000 moveto 
120.000  82.000 lineto 
stroke 
 65.000  55.000 moveto 
 85.000 105.000 lineto 
stroke 
[   2.000   1.000 ] 0 setdash 
 80.000  77.625 moveto 
 65.000  77.625 lineto 
stroke 
 84.332  70.000 moveto 
 84.332  55.000 lineto 
stroke 
 50.000  51.031 moveto 
 65.000  51.031 lineto 
stroke 
 68.892  56.500 moveto
 68.892  80.000 lineto 
stroke 
 72.608  53.500 moveto 
 72.608  37.000 lineto 
stroke 
 80.000  47.450 moveto 
 65.000  47.450 lineto 
stroke
 65.000  47.450 moveto 
 45.000 rotmem 
 51.000   1.000 moveto
  1.000 nbpt mul symb  (l = ) show currentpoint 
  1.000 nbpt mul rm  (0) show currentpoint 
clear 
rotrev 
 25.000 rotmem 
 51.200  2.800 moveto 
  1.000 nbpt mul it  (y = e) show currentpoint 
  0.035 nbpt mul   0.175 nbpt mul rmoveto 
  0.700 nbpt mul symb  (l) show <A2> show currentpoint 
1 index 3 index moveto 
  1.000 nbpt mul it  (x) show currentpoint 
clear 
rotrev 
 70.000  20.000 moveto 
 90.000 rotmem 
 -2.000  -3.500 moveto 
  1.000 nbpt mul it  (x ) show currentpoint 
  1.000 nbpt mul rm  (= ) show currentpoint 
  1.000 nbpt mul symb  (-) show currentpoint 
  1.000 nbpt mul rm  ( cos ) show currentpoint 
  1.000 nbpt mul symb  (l) show <A2> show currentpoint 
clear 
 62.000   0.300 moveto 
  1.000 nbpt mul it  (x) show currentpoint 
  1.000 nbpt mul rm  ( = cos ) show currentpoint 
  1.000 nbpt mul symb  (l) show currentpoint 
clear 
rotrev 
 74.500  81.500 moveto 
 69.000 rotmem 
  1.000 nbpt mul it  (y = e) show currentpoint 
  0.035 nbpt mul   0.175 nbpt mul rmoveto 
  0.700 nbpt mul symb  (l) show currentpoint 
1 index 3 index moveto 
  1.000 nbpt mul it  (x) show currentpoint 
clear 
grestore}
\LabelTeX   61.300  56.500 $0$\ELTX
\LabelTeX   44.000  56.500 $-1$\ELTX
\LabelTeX   80.700  56.500 $1$\ELTX
\LabelTeX   85.500  56.500 $\cosh\,\lambda'$\ELTX
\LabelTeX   30.300  49.500 $y=-\cos\,\lambda$\ELTX
\LabelTeX   81.300  46.800 $y=\cos\,\lambda'$\ELTX
\LabelTeX   56.500  76.500 $\cosh\,\lambda$\ELTX
\LabelTeX  117.200  50.200 $x$\ELTX
\LabelTeX   60.500  96.600 $y$\ELTX
\LabelTeX   61.700  71.500 $1$\ELTX
\LabelTeX   58.800  35.500 $-1$\ELTX
\EndFig
\vskip-2mm

\centerline{On a figur\'e ici $\lambda>0$, $\lambda'<0$.}

\subsection 1.3. Cas o\`u l'on conna\^{\i}t une \og$\,$int\'egrale 
premi\`ere$\,$\fg|

\noindent Supposons qu'on cherche \`a r\'esoudre une \'equation
$$
y'=f(x,y)\leqno{\rm (E)}
$$
ou un syst\`eme diff\'erentiel
$$
\left\{\eqalign{
\dfrac{dx}{dt}=a(x,y)&\cr
\noalign{\vskip4pt}
\dfrac{dy}{dt}=b(x,y)&\cr
}\right.\leqno{\rm (S)}
$$
dans un ouvert $U\subset \bR^2$. Dans les deux cas on a une \'ecriture
sous forme diff\'erentielle~:
$$
\eqalign{
&{\rm (E)}\Leftrightarrow f(x,y)dx-dy=0,\cr
\noalign{\vskip4pt}
&{\rm (S)}\Rightarrow b(x,y)dx-a(x,y)dy=0.\cr
}
$$

\bigskip
\majdeb
\state D\'efinition -- |  {\it On dit qu'une fonction $V:U\rightarrow\bR$ de 
classe $C^1$ est une int\'egrale premi\`ere si (E) (respectivement (S))
implique
$$
dV=V'_x(x,y)dx+V'_y(x,y)dy=0.
$$}
\vskip-\baselineskip
\majfin
\vskip\baselineskip

\noindent Dans ce cas, les courbes int\'egrales $y=\varphi(x)$ v\'erifient 
$$
V'_x(x,\varphi(x))+V'_y(x,\varphi(x))\varphi'(x)=\frac{d}{dx}\ [V(x,\varphi(x)]=0.
$$
Les courbes int\'egrales sont donc contenues dans les lignes de niveau
$V(x,y)=\lambda$, o\`u $\lambda\in\bR$ est une constante.

\InsertFig 41.000 43.000 {
1 mm unit
gsave 
  0.000   0.000 moveto 
 70.000  35.000 rect  clip  newpath 
 10.000  20.000 moveto 
[  10.000  20.000    11.053  20.128    11.468  20.171    11.886  20.209   
 12.305  20.244    12.726  20.274    13.148  20.299    13.570  20.319   
 13.994  20.335    14.418  20.345    14.843  20.350    15.206  20.349   
 15.545  20.345    15.885  20.338    16.224  20.326    16.562  20.311   
 16.900  20.292    17.237  20.268    17.574  20.241    17.909  20.210   
 18.243  20.174    18.576  20.134    18.907  20.090    19.237  20.041   
 19.565  19.988    19.891  19.931    20.215  19.868    20.536  19.802   
 20.856  19.730    21.172  19.655    21.486  19.574    21.797  19.489   
 22.105  19.399    22.410  19.304    22.711  19.204    23.009  19.100   
 23.304  18.991    23.594  18.877    23.880  18.758    24.163  18.635   
 24.441  18.507    24.714  18.374    24.983  18.236    25.247  18.094   
 25.507  17.947    25.761  17.796    26.010  17.640    26.254  17.479   
 26.492  17.314    26.725  17.145    26.952  16.971    27.173  16.793   
 27.388  16.611    27.597  16.425    27.800  16.235    27.997  16.041   
 28.187  15.843    28.370  15.642    28.547  15.437    28.717  15.228   
 28.881  15.016    29.037  14.801    29.187  14.583    29.329  14.362   
 29.464  14.138    29.592  13.911    29.713  13.682    29.826  13.450   
 29.932  13.216    30.030  12.980    30.121  12.741    30.204  12.501   
 30.279  12.260    30.347  12.016    30.407  11.772    30.459  11.526   
 30.503  11.279    30.540  11.031    30.569  10.783    30.590  10.534   
 30.603  10.285    30.608  10.035    30.605   9.786    30.595   9.536   
 30.576   9.287    30.550   9.039    30.516   8.791    30.474   8.544   
 30.424   8.298    30.366   8.053    30.301   7.809    30.228   7.567   
 30.147   7.326    30.059   7.087    29.963   6.850    29.860   6.616   
 29.746   6.378    29.597   6.091    29.425   5.788    29.202   5.433   
 28.884   4.980    28.345   4.322    27.606   3.575    26.776   2.887   
 25.863   2.261    24.873   1.701    23.993   1.285    23.225   0.974   
 22.497   0.719    21.859   0.524    21.250   0.360    20.630   0.214   
 20.104   0.105    19.596   0.012    19.082  -0.069    18.565  -0.140   
] scurve 
 10.000  20.000 moveto 
[  10.000  20.000     8.949  19.846     8.449  19.764     7.953  19.678   
  7.462  19.587     6.977  19.493     6.496  19.394     6.021  19.293   
  5.551  19.188     5.087  19.080     4.629  18.970     4.177  18.857   
  3.731  18.742     3.291  18.626     2.858  18.508     2.431  18.388   
  2.010  18.268     1.595  18.146     1.187  18.024     0.786  17.902   
  0.391  17.779     0.002  17.655    -0.380  17.532   ] scurve 
 15.000  30.000 moveto 
[  15.000  30.000    16.205  29.989    16.785  29.976    17.364  29.957   
 17.942  29.933    18.519  29.903    19.094  29.867    19.669  29.825   
 20.242  29.776    20.813  29.721    21.381  29.659    21.948  29.591   
 22.512  29.515    23.073  29.432    23.631  29.343    24.185  29.245   
 24.736  29.141    25.283  29.028    25.826  28.908    26.365  28.780   
 26.899  28.644    27.428  28.501    27.952  28.349    28.471  28.188   
 28.984  28.020    29.491  27.843    29.991  27.658    30.486  27.465   
 30.973  27.263    31.454  27.053    31.928  26.835    32.394  26.608   
 32.852  26.372    33.303  26.128    33.746  25.876    34.180  25.616   
 34.606  25.347    35.023  25.070    35.431  24.785    35.831  24.492   
 36.221  24.191    36.601  23.882    36.972  23.566    37.333  23.242   
 37.685  22.910    38.026  22.571    38.357  22.225    38.678  21.872   
 38.988  21.513    39.288  21.146    39.578  20.773    39.856  20.394   
 40.124  20.008    40.381  19.617    40.626  19.220    40.861  18.817   
 41.085  18.409    41.297  17.996    41.499  17.579    41.689  17.157   
 41.867  16.730    42.035  16.300    42.191  15.865    42.335  15.427   
 42.468  14.986    42.590  14.542    42.700  14.095    42.798  13.645   
 42.885  13.194    42.960  12.740    43.024  12.285    43.076  11.828   
 43.117  11.370    43.146  10.911    43.163  10.452    43.169   9.993   
 43.163   9.533    43.146   9.074    43.117   8.615    43.076   8.157   
 43.024   7.700    42.960   7.245    42.885   6.791    42.798   6.340   
 42.699   5.890   ] scurve
 15.000  30.000 moveto 
[  15.000  30.000    13.798  29.990    13.220  29.978    12.641  29.962   
 12.064  29.941    11.487  29.917    10.911  29.889    10.336  29.858   
  9.763  29.823     9.190  29.785     8.620  29.744     8.050  29.701   
  7.483  29.654     6.917  29.606     6.352  29.555     5.790  29.502   
  5.230  29.448     4.671  29.392     4.115  29.334     3.560  29.275   
  3.008  29.215     2.458  29.154     1.910  29.092     1.365  29.030   
  0.821  28.967     0.280  28.904    -0.259  28.841    -0.796  28.778   
] scurve 
 19.000  10.000 moveto 
[  19.000  10.000    18.580   8.888    18.167   8.494    17.827   8.264   
 17.585   8.132    17.378   8.036    17.187   7.958    17.020   7.898   
 16.848   7.842    16.701   7.800    16.558   7.763    16.413   7.730   
 16.265   7.700    16.146   7.678    16.026   7.659    15.904   7.642   
 15.782   7.627    15.659   7.614    15.543   7.604    15.444   7.597   
 15.344   7.592    15.245   7.588    15.145   7.585    15.045   7.584   
 14.945   7.584    14.846   7.585    14.746   7.588    14.646   7.592   
 14.547   7.598    14.448   7.605    14.349   7.613    14.251   7.622   
 14.153   7.633    14.055   7.645    13.958   7.658    13.862   7.672   
 13.766   7.688    13.671   7.705    13.576   7.723    13.483   7.742   
 13.390   7.762    13.298   7.784    13.207   7.806    13.117   7.830   
 13.028   7.855    12.940   7.881    12.853   7.908    12.767   7.935   
 12.682   7.964    12.599   7.994    12.517   8.025    12.436   8.057   
 12.356   8.089    12.278   8.123    12.201   8.157    12.126   8.193   
 12.052   8.229    11.980   8.266    11.909   8.303    11.840   8.342   
 11.772   8.381    11.706   8.421    11.642   8.461    11.579   8.503   
 11.518   8.545    11.459   8.587    11.401   8.630    11.346   8.674   
 11.292   8.718    11.240   8.763    11.190   8.809    11.142   8.855   
 11.095   8.901    11.051   8.948    11.009   8.995    10.968   9.043   
 10.930   9.091    10.893   9.140    10.859   9.188    10.826   9.238   
 10.796   9.287    10.767   9.337    10.741   9.387    10.717   9.437   
 10.695   9.488    10.674   9.538    10.656   9.589    10.640   9.640   
 10.627   9.692    10.615   9.743    10.605   9.794    10.598   9.846   
 10.593   9.898    10.590   9.949    10.588  10.001    10.590  10.053   
 10.593  10.104    10.598  10.156    10.605  10.207    10.615  10.259   
 10.627  10.310    10.641  10.361    10.656  10.412    10.674  10.463   
 10.695  10.514    10.717  10.565    10.741  10.615    10.767  10.665   
 10.797  10.717    10.836  10.779    10.889  10.856    10.972  10.963   
 11.261  11.256    11.654  11.548    11.906  11.696    12.127  11.809   
 12.313  11.894    12.501  11.970    12.662  12.030    12.828  12.086   
 12.995  12.137    13.135  12.176    13.276  12.212    13.420  12.245   
 13.566  12.276    13.697  12.301    13.817  12.321    13.937  12.340   
 14.058  12.357    14.180  12.371    14.303  12.384    14.422  12.394   
 14.521  12.401    14.620  12.407    14.720  12.412    14.820  12.415   
 14.919  12.417    15.019  12.417    15.119  12.416    15.219  12.414   
 15.318  12.410    15.418  12.405    15.517  12.398    15.616  12.390   
 15.715  12.381    15.813  12.370    15.910  12.358    16.007  12.345   
 16.104  12.330    16.200  12.313    16.295  12.295    16.389  12.276   
 16.483  12.255    16.576  12.233    16.667  12.210    16.758  12.185   
 16.848  12.159    16.936  12.131    17.024  12.102    17.110  12.072   
 17.195  12.040    17.279  12.007    17.361  11.973    17.441  11.937   
 17.521  11.900    17.598  11.862    17.674  11.823    17.749  11.782   
 17.821  11.740    17.892  11.697    17.962  11.653    18.029  11.608   
 18.094  11.562    18.158  11.514    18.219  11.466    18.279  11.417   
 18.336  11.366    18.392  11.315    18.445  11.262    18.496  11.209   
 18.545  11.155    18.591  11.100    18.635  11.045    18.677  10.989   
 18.717  10.932    18.754  10.874    18.789  10.816    18.821  10.757   
 18.851  10.697    18.879  10.637   ] scurve 
[  18.879  10.637    18.904  10.577    18.927  10.516    18.947  10.455   
 18.964  10.394    18.979  10.332    18.992  10.270    19.002  10.208   
 19.009  10.145    19.014  10.083    19.016  10.020    19.015   9.958   
 19.012   9.895    19.007   9.833    18.999   9.771    18.988   9.709   
 18.975   9.647    18.959   9.585    18.941   9.524    18.920   9.463   
 18.896   9.402    18.871   9.342    18.842   9.282    18.811   9.223   
 18.777   9.163    18.732   9.090    18.671   9.001    18.569   8.871   
 18.159   8.485    17.824   8.260    17.581   8.128    17.379   8.034   
 17.188   7.956    17.020   7.895    16.848   7.840    16.701   7.798   
 16.558   7.761    16.412   7.728    16.265   7.698    16.145   7.676   
 16.025   7.657    15.903   7.640    15.781   7.625    15.658   7.612   
 15.544   7.603    15.445   7.596    15.345   7.590    15.246   7.586   
 15.146   7.583    15.046   7.582    14.946   7.582    14.846   7.584   
 14.746   7.586    14.647   7.590    14.547   7.596   ] scurve 
 19.000  10.000 moveto 
[  19.000  10.000    18.595  11.107    18.513  11.202    18.462  11.255   
 18.409  11.308    18.355  11.360    18.298  11.410    18.238  11.460   
 18.177  11.509    18.114  11.557    18.049  11.603    17.982  11.649   
 17.913  11.693    17.843  11.737    17.770  11.779    17.696  11.820   
 17.620  11.859    17.543  11.898    17.464  11.935    17.383  11.971   
 17.302  12.006    17.218  12.039    17.133  12.071    17.047  12.102   
 16.960  12.131    16.872  12.159    16.782  12.186    16.692  12.211   
 16.600  12.235   ] scurve 
  1.000  10.000 moveto 
[   1.000  10.000     1.255  11.148     1.376  11.384     1.519  11.621   
  1.685  11.856     1.908  12.131     2.541  12.763     3.397  13.424   
  4.057  13.847     4.637  14.176     5.155  14.442     5.664  14.683   
  6.125  14.885     6.605  15.080     7.048  15.249     7.459  15.395   
  7.881  15.536     8.314  15.671     8.741  15.795     9.103  15.894   
  9.471  15.988     9.844  16.077    10.224  16.162    10.608  16.241   
 10.983  16.312    11.297  16.366    11.614  16.417    11.934  16.464   
 12.256  16.507    12.580  16.545    12.906  16.579    13.233  16.609   
 13.526  16.631    13.790  16.648    14.055  16.661    14.320  16.671   
 14.585  16.678    14.850  16.682    15.116  16.682    15.382  16.679   
 15.647  16.672    15.912  16.662    16.176  16.648    16.440  16.630   
 16.703  16.608    16.965  16.583    17.226  16.554    17.486  16.521   
 17.744  16.484    18.001  16.443    18.256  16.398    18.509  16.349   
 18.760  16.296    19.008  16.238    19.254  16.177    19.498  16.112   
 19.739  16.042    19.977  15.969    20.212  15.891    20.444  15.810   
 20.672  15.724    20.897  15.634    21.118  15.540    21.335  15.442   
 21.548  15.340    21.757  15.234    21.962  15.124    22.162  15.011   
 22.357  14.893    22.548  14.772    22.734  14.647    22.914  14.519   
 23.090  14.387    23.260  14.251    23.425  14.112    23.584  13.970   
 23.737  13.825    23.885  13.676    24.026  13.525    24.162  13.371   
 24.291  13.213    24.414  13.054    24.531  12.891    24.641  12.726   
 24.745  12.559    24.842  12.390    24.932  12.219    25.016  12.046   
 25.093  11.871    25.162  11.695    25.225  11.517    25.281  11.337   
 25.330  11.157    25.372  10.976    25.407  10.794    25.434  10.611   
 25.455  10.427    25.468  10.243    25.474  10.059    25.473   9.875   
 25.464   9.691    25.449   9.508    25.426   9.324    25.396   9.142   
 25.359   8.960    25.315   8.779    25.264   8.599    25.206   8.420   
 25.141   8.243    25.068   8.067    24.989   7.892    24.904   7.720   
 24.811   7.549    24.688   7.343    24.531   7.103    24.289   6.779   
 23.779   6.211    23.004   5.543    22.161   4.984    21.492   4.628   
 20.937   4.379    20.434   4.183    19.978   4.028    19.527   3.893   
 19.146   3.792    18.758   3.701    18.365   3.619    18.018   3.557   
 17.697   3.506    17.373   3.461    17.047   3.422    16.720   3.390   
 16.403   3.364    16.139   3.347    15.874   3.333    15.609   3.323   
 15.343   3.317    15.078   3.314    14.812   3.315    14.546   3.319   
 14.281   3.326    14.016   3.337    13.751   3.351    13.488   3.368   
 13.224   3.389    12.962   3.412    12.701   3.438    12.441   3.467   
 12.182   3.499    11.925   3.534    11.668   3.571    11.414   3.611   
 11.161   3.653    10.910   3.698    10.661   3.745    10.414   3.795   
 10.169   3.846     9.926   3.900     9.685   3.956     9.447   4.014   
  9.211   4.074     8.978   4.136     8.747   4.199     8.518   4.265   
  8.293   4.332     8.070   4.400     7.850   4.470     7.632   4.542   
  7.418   4.615     7.207   4.690     6.998   4.766     6.793   4.843   
  6.591   4.921     6.392   5.000     6.196   5.081     6.004   5.162   
  5.815   5.245     5.629   5.328     5.446   5.413     5.267   5.498   
  5.091   5.584     4.919   5.671     4.750   5.758     4.585   5.846   
  4.423   5.935     4.265   6.025     4.111   6.115     3.960   6.206   
  3.812   6.297     3.669   6.388   ] scurve 
[   3.669   6.388     3.528   6.480     3.392   6.573     3.259   6.666   
  3.130   6.759     3.005   6.853     2.883   6.946     2.765   7.041   
  2.651   7.135     2.540   7.230     2.433   7.325     2.330   7.420   
  2.231   7.516     2.135   7.611     2.044   7.707     1.956   7.803   
  1.871   7.899     1.791   7.995     1.714   8.092     1.641   8.188   
  1.572   8.285     1.507   8.381     1.446   8.478     1.388   8.575   
  1.334   8.672     1.284   8.769     1.237   8.866     1.195   8.963   
  1.156   9.060     1.121   9.157     1.090   9.254     1.063   9.351   
  1.039   9.449     1.020   9.546     1.004   9.643     0.992   9.740   
  0.983   9.838     0.979   9.935     0.978  10.032     0.981  10.129   
  0.988  10.227     0.999  10.324     1.013  10.421     1.032  10.518   
  1.054  10.616     1.080  10.713     1.109  10.810     1.143  10.907   
  1.180  11.004     1.221  11.101     1.266  11.198     1.315  11.295   
  1.367  11.392     1.437  11.511     1.523  11.646     1.641  11.814   
  1.828  12.052     2.354  12.605     3.201  13.295     3.923  13.774   
  4.516  14.118     5.027  14.386     5.549  14.637     6.006  14.840   
  6.482  15.037     6.940  15.214     7.348  15.362     7.767  15.504   
  8.198  15.641     8.639  15.772     8.999  15.872     9.366  15.967   
  9.738  16.058    10.116  16.144    10.499  16.224    10.887  16.299   
 11.208  16.356    11.525  16.408    11.844  16.456    12.166  16.500   
 12.489  16.540    12.815  16.575    13.142  16.606    13.453  16.630   
] scurve 
  1.000  10.000 moveto 
[   1.000  10.000     1.251   8.852     1.312   8.727     1.364   8.630   
  1.419   8.533     1.478   8.437     1.541   8.340     1.608   8.243   
  1.679   8.147     1.753   8.050     1.831   7.954     1.913   7.858   
  1.998   7.762     2.088   7.666     2.181   7.570     2.278   7.475   
  2.379   7.379     2.483   7.284     2.591   7.189     2.703   7.095   
  2.819   7.000     2.938   6.906     3.061   6.813     3.188   6.719   
  3.318   6.626     3.452   6.533     3.590   6.441     3.731   6.349   
] scurve 
 15.000  25.000 moveto 
[  15.000  25.000    16.174  24.986    16.637  24.972    17.100  24.953   
 17.562  24.929    18.023  24.900    18.483  24.865    18.941  24.826   
 19.398  24.781    19.853  24.730    20.307  24.673    20.758  24.611   
 21.207  24.543    21.654  24.468    22.098  24.388    22.538  24.301   
 22.976  24.208    23.411  24.109    23.841  24.004    24.268  23.891   
 24.691  23.773    25.110  23.647    25.524  23.516    25.934  23.377   
 26.339  23.232    26.739  23.080    27.133  22.921    27.522  22.755   
 27.906  22.583    28.283  22.404    28.655  22.219    29.020  22.026   
 29.378  21.827    29.730  21.622    30.075  21.409    30.413  21.191   
 30.744  20.966    31.068  20.734    31.383  20.496    31.692  20.252   
 31.992  20.002    32.284  19.746    32.568  19.484    32.844  19.216   
 33.111  18.942    33.370  18.663    33.620  18.379    33.861  18.089   
 34.093  17.795    34.316  17.495    34.530  17.190    34.735  16.881   
 34.930  16.568    35.116  16.250    35.293  15.929    35.460  15.603   
 35.617  15.274    35.764  14.942    35.902  14.606    36.029  14.267   
 36.147  13.925    36.255  13.581    36.353  13.234    36.441  12.886   
 36.519  12.535    36.586  12.183    36.644  11.829    36.691  11.474   
 36.728  11.118    36.755  10.761    36.772  10.404    36.779  10.047   
 36.775   9.689    36.761   9.332    36.737   8.975    36.703   8.619   
 36.658   8.263    36.604   7.909    36.539   7.556    36.464   7.205   
 36.379   6.856    36.284   6.509    36.180   6.164    36.065   5.821   
 35.940   5.482    35.805   5.145    35.660   4.811    35.477   4.421   
 35.268   4.015    35.008   3.551    34.683   3.027    34.220   2.364   
 33.527   1.503    32.727   0.658    31.848  -0.133   ] scurve 
 15.000  25.000 moveto 
[  15.000  25.000    13.799  24.986    13.221  24.969    12.643  24.946   
 12.067  24.917    11.492  24.883    10.919  24.844    10.348  24.799   
  9.778  24.750     9.211  24.697     8.647  24.639     8.085  24.578   
  7.525  24.513     6.969  24.444     6.416  24.373     5.866  24.299   
  5.319  24.222     4.776  24.143     4.236  24.062     3.700  23.979   
  3.167  23.894     2.638  23.809     2.113  23.722     1.591  23.634   
  1.074  23.545     0.560  23.456     0.050  23.367    -0.456  23.278   
] scurve 
 15.000  10.000 moveto 
 31.000  30.000 moveto 
 47.000  13.000 lineto 
stroke 
 39.000  21.800 moveto 
  0.700 disk 
  0.000   0.000 rlineto 
stroke 
 39.000  21.800 moveto 
 13.454  43.264 vector 
 40.700  23.400 moveto 
 42.300  21.700 lineto
stroke
 42.300  21.700 moveto
 40.600  20.100 lineto 
stroke 
grestore}
\LabelTeX   51.400  28.100 $\overrightarrow{\grad}\,V$\ELTX
\LabelTeX   38.400   1.100 $V(x,y)=\lambda$\ELTX
\EndFig
\vskip8mm

\noindent En tout point o\`u
$\overrightarrow{\grad}\,V\not=\overrightarrow 0$, la ligne de niveau
correspondante poss\`ede une tangente perpendiculaire \`a
$\overrightarrow{\grad}\,V$. Le champ des tangentes est dirig\'e par le
vecteur
$$
\vec k\pmatrix{
1\cr
f(x,y)\cr
},\quad{\rm resp.}\quad \vec k\pmatrix{
a(x,y)\cr
b(x,y)\cr
}\quad\hbox{dans le cas de (E) (resp. (S)).}
$$
La condition d'orthogonalit\'e $\overrightarrow{\grad}\,V\perp\vec k$
\'equivaut \`a la proportionnalit\'e de l'\'equation $V'_xdx+V'_ydy=0$
\`a l'\'equation diff\'erentielle (E) (ou (S)). On peut donc \'enoncer~:

\bigskip
\majdeb
\state Propri\'et\'e caract\'eristique -- |  {\it $V$ est une
int\'egrale premi\`ere si et seulement si $\overrightarrow{\grad}\,V$
est orthogonal au champ des tangentes de l'\'equation diff\'erentielle
consid\'er\'ee.}  \par\majfin\par

\bigskip
\state Exemple -- |   Soit $y'=\dfrac{y}{x+y^2}$\quad sur\quad
$U=\{x+y^2\not=0\}$. L'\'equation se r\'ecrit 
$$
ydx-(x+y^2)dy=0.\leqno{\rm (E)}
$$
Cette diff\'erentielle n'est pas une diff\'erentielle exacte
$dV=Pdx+Qdy$ (on devrait avoir $\dfrac{\partial P}{\partial
y}=\dfrac{\partial Q}{\partial x}$, ce qui n'est pas le cas). On
observe n\'eanmoins que
$$
d\Big(\frac{x}{y}\Big)=\frac{ydx-xdy}{y^2}.
$$
Multiplions alors l'\'equation (E) par $\dfrac{1}{y^2}$, en se pla\c{c}ant
dans l'ouvert $y\neq 0$~:
$$
{\rm (E)}\Leftrightarrow\frac{ydx-xdy}{y^2}-dy=0\Leftrightarrow
d\Big(\frac{x}{y}-y\Big)=0.
$$
Les courbes int\'egrales y sont donc donn\'ees par
$$
\frac{x}{y}-y=\lambda\quad\Leftrightarrow\quad x=y^2+\lambda y.
$$
Ce sont des arcs de la parabole d'axe $y=-\dfrac{\lambda}{2}$ et de
sommet $\pmatrix{-\lambda^2/4\cr -\lambda/2\cr}$, d\'elimit\'es par
les points tels que $x+y^2=2y^2+\lambda y=0$,  c'est-\`a-dire
$\pmatrix{0\cr 0\cr}$ et le sommet, qui doivent \^etre exclus. Par
ailleurs, $y=0$ est une solution singuli\`ere, fournissant deux
solutions maximales pour $x\in{}]-\infty,0[$ et $x\in{}]0,+\infty[$
respectivement.

\bigskip
\state Remarque -- |  On dit que $\dfrac{1}{y^2}$ est un \og$\,$facteur
int\'egrant$\,$\fg\ de la forme diff\'erentielle $ydx-(x+y^2)dy=0$.

\InsertFig 16.000 82.000 {
1 mm unit
  0.000  35.000 moveto 
100.000   0.000 vector 
 50.000   0.000 moveto 
 70.000  90.000 vector 
gsave 
  5.000   5.000 moveto 
 90.000  60.000 rect  clip  newpath 
 50.000  35.000 moveto 
 50.000  55.000 moveto 
[  50.000  55.000    48.555  53.850    47.355  52.766    46.366  51.744   
 45.562  50.781    44.916  49.873    44.458  49.109    44.161  48.525   
 43.945  48.031    43.798  47.642    43.674  47.264    43.586  46.950   
 43.517  46.663    43.460  46.384    43.416  46.126    43.387  45.912   
 43.365  45.702    43.348  45.496    43.338  45.294    43.332  45.107   
 43.332  44.951    43.334  44.798    43.340  44.647    43.348  44.498   
 43.360  44.352    43.373  44.208    43.390  44.066    43.408  43.926   
 43.429  43.789    43.452  43.653    43.478  43.520    43.505  43.389   
 43.533  43.259    43.564  43.132    43.596  43.007    43.630  42.883   
 43.665  42.762    43.702  42.642    43.740  42.524    43.779  42.408   
 43.819  42.294    43.861  42.182    43.903  42.071    43.946  41.962   
 43.991  41.855    44.036  41.749    44.082  41.645    44.128  41.543   
 44.175  41.442    44.223  41.343    44.271  41.245    44.320  41.149   
 44.369  41.054    44.419  40.961    44.469  40.869    44.519  40.778   
 44.570  40.689    44.621  40.601    44.672  40.515    44.723  40.430   
 44.775  40.346    44.826  40.264    44.878  40.183    44.930  40.103   
 44.981  40.024    45.033  39.947    45.085  39.871    45.137  39.796   
 45.188  39.722    45.240  39.649    45.291  39.577    45.343  39.507   
 45.394  39.437    45.445  39.369    45.496  39.301    45.547  39.235   
 45.597  39.170    45.647  39.106    45.697  39.042    45.747  38.980   
 45.796  38.919    45.846  38.858    45.895  38.799    45.943  38.740   
 45.992  38.683    46.040  38.626    46.087  38.570    46.135  38.515   
 46.182  38.461    46.228  38.407    46.275  38.355    46.321  38.303   
 46.366  38.252    46.412  38.202    46.457  38.153    46.501  38.104   
 46.545  38.056    46.589  38.009    46.633  37.963    46.676  37.917   
 46.718  37.872    46.760  37.828    46.802  37.784    46.844  37.741   
 46.885  37.699    46.925  37.658    46.965  37.617    47.005  37.576   
 47.045  37.537    47.084  37.497    47.122  37.459    47.161  37.421   
 47.198  37.384    47.236  37.347    47.273  37.311    47.309  37.275   
 47.346  37.240    47.381  37.206    47.417  37.172    47.452  37.138   
 47.487  37.105    47.521  37.073    47.555  37.041    47.588  37.009   
 47.621  36.978    47.654  36.948    47.686  36.918    47.718  36.888   
 47.750  36.859    47.781  36.830    47.812  36.802    47.842  36.774   
 47.872  36.747    47.902  36.720    47.931  36.694    47.960  36.668   
 47.989  36.642    48.017  36.617    48.045  36.592    48.072  36.567   
 48.100  36.543    48.126  36.519    48.153  36.496    48.179  36.473   
 48.205  36.450    48.231  36.428    48.256  36.406    48.281  36.384   
 48.305  36.363    48.329  36.342    48.353  36.321    48.377  36.301   
 48.400  36.281    48.423  36.261    48.446  36.241    48.468  36.222   
 48.490  36.203    48.512  36.185    48.534  36.167    48.555  36.149   
 48.576  36.131    48.596  36.113    48.617  36.096    48.637  36.079   
 48.657  36.063    48.676  36.046    48.696  36.030    48.715  36.014   
 48.733  35.999    48.752  35.983    48.770  35.968    48.788  35.953   
 48.806  35.939    48.823  35.924    48.841  35.910    48.858  35.896   
 48.874  35.882    48.891  35.868    48.907  35.855    48.923  35.842   
 48.939  35.829    48.955  35.816    48.970  35.804    48.986  35.791   
 49.001  35.779    49.015  35.767    49.030  35.755    49.044  35.743   
 49.058  35.732    49.072  35.721   ] scurve 
[  49.072  35.721    49.086  35.710    49.100  35.699    49.113  35.688   
 49.126  35.677    49.139  35.667    49.152  35.657    49.165  35.646   
 49.177  35.636    49.189  35.627    49.202  35.617    49.213  35.607   
 49.225  35.598    49.237  35.589    49.248  35.580    49.259  35.571   
 49.271  35.562    49.281  35.553    49.292  35.545    49.303  35.537   
 49.313  35.528    49.324  35.520    49.334  35.512    49.344  35.504   
 49.354  35.496    49.363  35.489    49.373  35.481    49.382  35.474   
 49.392  35.466    49.401  35.459    49.410  35.452    49.419  35.445   
 49.427  35.438    49.436  35.432    49.445  35.425    50.000  35.000  ] scurve 
 50.000  55.000 moveto 
[  50.000  55.000    51.694  56.198    53.239  57.189    54.682  58.046   
 56.072  58.822    57.461  59.556    58.825  60.242    60.112  60.863   
 61.483  61.498    62.944  62.149    64.180  62.681    65.480  63.224   
 66.845  63.777    68.280  64.342    69.786  64.917    71.044  65.385   
] scurve 
 50.000  15.000 moveto 
[  50.000  15.000    49.353  15.497    49.111  15.690    48.877  15.881   
 48.649  16.070    48.428  16.257    48.214  16.442    48.007  16.626   
 47.806  16.808    47.612  16.988    47.424  17.166    47.241  17.342   
 47.065  17.517    46.894  17.690    46.729  17.861    46.570  18.030   
 46.416  18.198    46.267  18.365    46.123  18.529    45.984  18.692   
 45.850  18.853    45.721  19.013    45.596  19.171    45.476  19.328   
 45.361  19.483    45.249  19.636    45.142  19.788    45.039  19.939   
 44.940  20.088    44.845  20.235    44.753  20.381    44.666  20.526   
 44.582  20.669    44.501  20.811    44.424  20.951    44.350  21.090   
 44.280  21.227    44.212  21.364    44.148  21.498    44.087  21.632   
 44.029  21.764    43.974  21.895    43.921  22.025    43.871  22.153   
 43.824  22.280    43.779  22.406    43.737  22.530    43.698  22.654   
 43.661  22.776    43.626  22.897    43.593  23.017    43.563  23.135   
 43.534  23.252    43.508  23.369    43.484  23.484    43.462  23.597   
 43.442  23.710    43.423  23.822    43.407  23.932    43.392  24.042   
 43.379  24.150    43.368  24.258    43.358  24.364    43.350  24.469   
 43.343  24.573    43.338  24.676    43.334  24.778    43.332  24.879   
 43.331  24.980    43.331  25.079    43.333  25.177    43.336  25.274   
 43.340  25.370    43.345  25.465    43.352  25.560    43.359  25.653   
 43.368  25.745    43.378  25.837    43.388  25.928    43.400  26.017   
 43.412  26.106    43.426  26.194    43.440  26.281    43.456  26.367   
 43.472  26.453    43.488  26.537    43.506  26.621    43.525  26.704   
 43.544  26.786    43.563  26.867    43.584  26.948    43.605  27.027   
 43.627  27.106    43.649  27.184    43.675  27.270    43.704  27.366   
 43.734  27.460    43.766  27.553    43.797  27.645    43.830  27.736   
 43.863  27.825    43.904  27.932    43.948  28.041    43.992  28.148   
 44.037  28.254    44.083  28.358    44.135  28.471    44.194  28.597   
 44.254  28.720    44.315  28.841    44.378  28.962    44.456  29.107   
 44.534  29.248    44.614  29.386    44.706  29.542    44.807  29.705   
 44.908  29.863    45.032  30.051    45.158  30.235    45.307  30.443   
 45.467  30.659    45.659  30.909    45.886  31.191    46.168  31.523   
 46.572  31.971    47.259  32.675    47.999  33.366    48.557  33.852   
 48.902  34.140    49.163  34.351    49.364  34.511    49.498  34.615   
 49.598  34.693    49.678  34.755    49.743  34.805    49.794  34.844   
 49.836  34.876    49.869  34.901    49.895  34.921    49.916  34.937   
 49.933  34.950    49.947  34.960    49.957  34.968    49.966  34.974   
 49.973  34.980    49.978  34.984    49.983  34.987    49.986  34.990   
 49.989  34.992    49.991  34.993    49.993  34.995    49.994  34.996   
 49.996  34.997    49.996  34.997    49.997  34.998    49.998  34.998   
 49.998  34.999    49.999  34.999    49.999  34.999    49.999  34.999   
 49.999  34.999    49.999  35.000   ] scurve 
 50.000  15.000 moveto 
[  50.000  15.000    50.694  14.492    50.983  14.287    51.281  14.080   
 51.588  13.872    51.904  13.661    52.229  13.448    52.563  13.233   
 52.907  13.015    53.261  12.796    53.625  12.574    53.999  12.350   
 54.384  12.124    54.779  11.896    55.185  11.665    55.603  11.432   
 56.032  11.197    56.473  10.959    56.926  10.719    57.391  10.477   
 57.869  10.232    58.360   9.985    58.864   9.735    59.381   9.483   
 59.912   9.228    60.347   9.023    60.792   8.815    61.245   8.606   
 61.708   8.396    62.181   8.184    62.664   7.970    63.156   7.754   
 63.659   7.537    64.172   7.317    64.696   7.097    65.231   6.874   
 65.776   6.650    66.333   6.423    66.901   6.195    67.480   5.966   
 68.072   5.734    68.675   5.500    69.290   5.265    69.918   5.028   
 70.558   4.789    71.212   4.548   ] scurve 
 65.000  50.000 moveto 
[  65.000  50.000    63.786  49.380    62.744  48.826    61.781  48.294   
 60.891  47.781    60.068  47.289    59.307  46.815    58.603  46.360   
 57.953  45.922    57.352  45.501    56.796  45.096    56.282  44.707   
 55.756  44.292    55.147  43.787    54.351  43.078    53.591  42.339   
 52.837  41.523    52.232  40.786    51.755  40.131    51.424  39.621   
 51.175  39.198    51.007  38.887    50.864  38.600    50.741  38.334   
 50.652  38.126    50.576  37.940    50.510  37.765    50.451  37.600   
 50.398  37.445    50.352  37.299    50.314  37.169    50.284  37.065   
 50.258  36.966    50.234  36.872    50.212  36.782    50.192  36.697   
 50.174  36.615    50.158  36.538    50.143  36.464    50.129  36.394   
 50.117  36.327    50.106  36.263    50.096  36.202    50.087  36.145   
 50.079  36.090    50.072  36.038    50.065  35.988    50.059  35.940   
 50.053  35.895    50.048  35.852    50.044  35.814    50.041  35.783   
 50.038  35.753    50.035  35.724    50.032  35.696    50.030  35.669   
 50.028  35.643    50.025  35.618    50.024  35.594    50.022  35.572   
 50.020  35.550    50.019  35.528    50.017  35.508    50.016  35.488   
 50.000  35.000  ] scurve 
 65.000  50.000 moveto 
[  65.000  50.000    66.323  50.647    67.458  51.182    68.592  51.699   
 69.799  52.233    71.085  52.784    72.250  53.269    73.399  53.735   
 74.608  54.212    75.878  54.702    77.215  55.204    78.542  55.691   
 79.715  56.112    80.936  56.542    82.207  56.980    83.531  57.427   
 84.909  57.883    86.344  58.349    87.838  58.824    89.077  59.211   
 90.357  59.604    91.678  60.004    93.043  60.410    94.453  60.822   
 95.909  61.242   ] scurve 
 65.000  20.000 moveto 
[  65.000  20.000    64.217  20.397    63.856  20.584    63.503  20.768   
 63.159  20.950    62.824  21.130    62.498  21.308    62.180  21.483   
 61.870  21.656    61.568  21.827    61.274  21.996    60.987  22.163   
 60.707  22.327    60.435  22.489    60.169  22.649    59.910  22.808   
 59.658  22.964    59.412  23.118    59.173  23.270    58.939  23.420   
 58.712  23.568    58.490  23.715    58.274  23.859    58.064  24.002   
 57.858  24.143    57.658  24.282    57.464  24.419    57.274  24.555   
 57.089  24.688    56.908  24.820    56.732  24.951    56.561  25.079   
 56.394  25.206    56.231  25.332    56.073  25.456    55.918  25.578   
 55.768  25.699    55.621  25.818    55.478  25.935    55.339  26.051   
 55.203  26.166    55.070  26.279    54.941  26.391    54.816  26.501   
 54.693  26.610    54.574  26.717    54.457  26.823    54.344  26.928   
 54.233  27.031    54.126  27.133    54.021  27.234    53.918  27.333   
 53.819  27.432    53.722  27.529    53.627  27.624    53.535  27.719   
 53.445  27.812    53.357  27.904    53.272  27.995    53.188  28.084   
 53.107  28.173    53.028  28.260    52.951  28.347    52.876  28.432   
 52.803  28.516    52.732  28.599    52.662  28.681    52.594  28.762   
 52.528  28.842    52.464  28.921    52.401  28.998    52.340  29.075   
 52.281  29.151    52.223  29.226    52.166  29.300    52.111  29.373   
 52.057  29.445    52.005  29.516    51.954  29.586    51.904  29.656   
 51.856  29.724    51.808  29.792    51.762  29.858    51.718  29.924   
 51.674  29.989    51.631  30.053    51.590  30.117    51.549  30.179   
 51.510  30.241    51.472  30.302    51.434  30.362    51.398  30.421   
 51.362  30.480    51.327  30.538    51.294  30.595    51.261  30.651   
 51.229  30.707    51.197  30.762    51.167  30.816    51.137  30.870   
 51.108  30.923    51.080  30.975    51.053  31.026    51.026  31.077   
 51.000  31.127    50.974  31.177    50.950  31.226    50.925  31.274   
 50.902  31.322    50.879  31.369    50.857  31.416    50.835  31.461   
 50.813  31.507    50.793  31.552    50.773  31.596    50.753  31.639   
 50.734  31.682    50.715  31.725    50.697  31.767    50.679  31.808   
 50.662  31.849    50.645  31.889    50.629  31.929    50.613  31.968   
 50.597  32.007    50.582  32.046    50.567  32.083    50.553  32.121   
 50.539  32.158    50.525  32.194    50.512  32.230    50.499  32.265   
 50.486  32.300    50.474  32.335    50.461  32.369    50.450  32.403   
 50.438  32.436    50.427  32.469    50.416  32.501    50.406  32.533   
 50.395  32.565    50.385  32.596    50.375  32.627    50.366  32.657   
 50.357  32.687    50.348  32.717    50.339  32.746    50.330  32.775   
 50.322  32.803    50.314  32.831    50.306  32.859    50.298  32.887   
 50.290  32.914    50.283  32.940    50.276  32.967    50.269  32.993   
 50.262  33.018    50.255  33.044    50.249  33.069    50.242  33.094   
 50.236  33.118    50.230  33.142    50.224  33.166    50.219  33.189   
 50.213  33.213    50.208  33.235    50.202  33.258    50.197  33.280   
 50.192  33.302    50.187  33.324    50.182  33.346    50.178  33.367   
 50.173  33.388    50.169  33.408    50.165  33.429    50.160  33.449   
 50.156  33.469    50.152  33.488    50.148  33.508    50.145  33.527   
 50.141  33.546    50.137  33.564    50.134  33.583    50.131  33.601   
 50.127  33.619    50.124  33.636    50.121  33.654    50.118  33.671   
 50.115  33.688    50.112  33.705   ] scurve 
[  50.112  33.705    50.109  33.721    50.106  33.738    50.104  33.754   
 50.000  35.000  ] scurve 
 65.000  20.000 moveto 
[  65.000  20.000    65.823  19.594    66.236  19.394    66.659  19.192   
 67.094  18.987    67.539  18.780    67.997  18.570    68.466  18.357   
 68.948  18.141    69.442  17.923    69.949  17.702    70.469  17.477   
 71.003  17.250    71.551  17.020    72.092  16.796    72.552  16.608   
 73.022  16.417    73.501  16.225    73.990  16.030    74.489  15.834   
 74.999  15.636    75.519  15.435    76.050  15.232    76.593  15.028   
 77.146  14.821    77.711  14.612    78.288  14.401    78.876  14.188   
 79.477  13.972    80.091  13.755    80.717  13.535    81.356  13.313   
 82.009  13.088    82.675  12.861    83.355  12.632    84.049  12.400   
 84.758  12.166    85.481  11.930    86.220  11.691    86.973  11.450   
 87.743  11.206    88.528  10.960    89.330  10.711    90.149  10.460   
 90.831  10.252    91.510  10.047    92.199   9.841    92.900   9.633   
 93.613   9.423    94.338   9.211    95.075   8.998    95.824   8.783   
] scurve 
 78.563  42.141 moveto 
[  78.563  42.141    77.395  41.879    76.526  41.683    75.687  41.493   
 74.876  41.308    74.093  41.128    73.336  40.953    72.605  40.784   
 71.898  40.619    71.215  40.459    70.554  40.303    69.916  40.152   
 69.299  40.005    68.702  39.863    68.124  39.724    67.566  39.589   
 67.026  39.459    66.503  39.331    65.998  39.208    65.509  39.088   
 65.035  38.972    64.577  38.858    64.133  38.748    63.704  38.642   
 63.289  38.538    62.886  38.437    62.497  38.339    62.120  38.244   
 61.754  38.151    61.401  38.062    61.058  37.974    60.726  37.890   
 60.405  37.807    60.093  37.727    59.791  37.649    59.499  37.574   
 59.216  37.501    58.941  37.429    58.675  37.360    58.417  37.293   
 58.167  37.227    57.925  37.164    57.690  37.102    57.463  37.042   
 57.242  36.984    57.028  36.928    56.821  36.873    56.620  36.819   
 56.425  36.767    56.235  36.717    56.052  36.668    55.874  36.620   
 55.702  36.574    55.534  36.529    55.372  36.486    55.215  36.443   
 55.062  36.402    54.914  36.362    54.770  36.324    54.631  36.286   
 54.496  36.249    54.365  36.214    54.237  36.179    54.114  36.145   
 53.994  36.113    53.878  36.081    53.765  36.050    53.655  36.020   
 53.549  35.991    53.446  35.963    53.346  35.935    53.249  35.909   
 53.155  35.883    53.063  35.858    52.974  35.833    52.888  35.809   
 52.805  35.786    52.723  35.764    52.645  35.742    52.568  35.721   
 52.494  35.701    52.422  35.681    52.352  35.661    52.284  35.642   
 52.218  35.624    52.154  35.606    52.092  35.589    52.031  35.572   
 51.973  35.556    51.916  35.540    51.861  35.525    51.807  35.510   
 51.755  35.495    51.705  35.481    51.656  35.467    51.608  35.454   
 51.562  35.441    51.517  35.428    51.473  35.416    51.431  35.404   
 51.390  35.393    51.350  35.382    51.311  35.371    51.273  35.360   
 51.237  35.350    51.201  35.340    51.167  35.330    51.133  35.321   
 51.101  35.312    51.069  35.303    51.038  35.294    51.009  35.286   
 50.980  35.278    50.952  35.270    50.924  35.262    50.898  35.255   
 50.872  35.247    50.847  35.240    50.823  35.233    50.799  35.227   
 50.776  35.220    50.754  35.214    50.732  35.208    50.711  35.202   
 50.691  35.196    50.671  35.191    50.652  35.185    50.633  35.180   
 50.615  35.175    50.598  35.170    50.580  35.165    50.564  35.160   
 50.548  35.156    50.532  35.151    50.517  35.147    50.502  35.143   
 50.488  35.139    50.000  35.000  ] scurve 
 78.563  42.141 moveto 
[  78.563  42.141    79.786  42.414    80.773  42.632    81.794  42.856   
 82.853  43.087    83.951  43.325    85.088  43.570    86.268  43.822   
 87.490  44.082    88.759  44.349    90.074  44.624    91.287  44.876   
 92.411  45.108    93.569  45.345    94.761  45.588    95.990  45.836   
] scurve 
 78.563  27.859 moveto 
[  78.563  27.859    77.555  28.086    76.991  28.213    76.440  28.338   
 75.901  28.460    75.374  28.581    74.859  28.699    74.355  28.814   
 73.861  28.928    73.379  29.040    72.907  29.149    72.445  29.257   
 71.993  29.363    71.551  29.466    71.118  29.568    70.695  29.668   
 70.281  29.766    69.876  29.862    69.479  29.957    69.091  30.049   
 68.711  30.140    68.339  30.230    67.974  30.317    67.618  30.403   
 67.269  30.488    66.928  30.571    66.593  30.652    66.266  30.732   
 65.945  30.811    65.631  30.888    65.324  30.963    65.023  31.038   
 64.728  31.110    64.439  31.182    64.157  31.252    63.880  31.321   
 63.609  31.389    63.343  31.455    63.083  31.520    62.828  31.584   
 62.579  31.647    62.334  31.709    62.095  31.769    61.860  31.829   
 61.630  31.887    61.405  31.944    61.184  32.000    60.968  32.056   
 60.756  32.110    60.549  32.163    60.345  32.215    60.146  32.266   
 59.951  32.316    59.760  32.366    59.572  32.414    59.388  32.462   
 59.208  32.508    59.032  32.554    58.859  32.599    58.689  32.643   
 58.523  32.687    58.360  32.729    58.200  32.771    58.044  32.812   
 57.890  32.852    57.740  32.892    57.592  32.930    57.448  32.968   
 57.306  33.006    57.167  33.042    57.031  33.078    56.897  33.114   
 56.766  33.148    56.638  33.182    56.512  33.216    56.389  33.249   
 56.268  33.281    56.149  33.312    56.033  33.343    55.919  33.374   
 55.807  33.404    55.697  33.433    55.590  33.462    55.484  33.490   
 55.381  33.518    55.279  33.545    55.180  33.572    55.082  33.598   
 54.987  33.624    54.893  33.649    54.801  33.674    54.710  33.699   
 54.622  33.722    54.535  33.746    54.450  33.769    54.366  33.792   
 54.285  33.814    54.204  33.836    54.125  33.857    54.048  33.878   
 53.972  33.899    53.898  33.919    53.825  33.939    53.753  33.958   
 53.683  33.977    53.614  33.996    53.547  34.015    53.480  34.033   
 53.415  34.051    53.351  34.068    53.289  34.085    53.228  34.102   
 53.167  34.119    53.108  34.135    53.050  34.151    52.993  34.166   
 52.938  34.182    52.883  34.197    52.829  34.211    52.776  34.226   
 52.725  34.240    52.674  34.254    52.624  34.268    52.575  34.281   
 52.528  34.294    52.481  34.307    52.434  34.320    52.389  34.333   
 52.345  34.345    52.301  34.357    52.258  34.369    52.216  34.380   
 52.175  34.392    52.135  34.403    52.095  34.414    52.056  34.425   
 52.018  34.435    51.981  34.446    51.944  34.456    51.908  34.466   
 51.873  34.476    51.838  34.485    51.804  34.495    51.770  34.504   
 51.738  34.513    51.705  34.522    51.674  34.531    51.643  34.540   
 51.612  34.548    51.583  34.556    51.553  34.564    51.525  34.572   
 51.496  34.580    51.469  34.588    51.442  34.596    51.415  34.603   
 51.389  34.610    51.363  34.618    51.338  34.625    51.313  34.631   
 51.289  34.638    51.265  34.645    51.242  34.651    51.219  34.658   
 51.196  34.664    51.174  34.670    51.152  34.676    51.131  34.682   
 51.110  34.688    51.090  34.694    51.070  34.699    51.050  34.705   
 51.030  34.710    51.011  34.716    50.993  34.721    50.974  34.726   
 50.956  34.731    50.939  34.736    50.921  34.741    50.904  34.746   
 50.888  34.750    50.871  34.755    50.855  34.759    50.840  34.764   
 50.824  34.768    50.809  34.773    50.794  34.777    50.779  34.781   
 50.765  34.785    50.751  34.789   ] scurve 
[  50.751  34.789    50.737  34.793    50.723  34.796    50.710  34.800   
 50.697  34.804    50.684  34.807    50.672  34.811    50.000  35.000  ] scurve 
 78.563  27.859 moveto 
[  78.563  27.859    79.609  27.627    80.232  27.489    80.869  27.349   
 81.520  27.206    82.187  27.061    82.869  26.912    83.566  26.761   
 84.280  26.607    85.010  26.451    85.758  26.291    86.522  26.129   
 87.305  25.963    88.106  25.794    88.926  25.622    89.766  25.447   
 90.625  25.269    91.505  25.087    92.405  24.902    93.327  24.713   
 94.150  24.546    94.920  24.390    95.704  24.232   ] scurve 
 35.719  56.422 moveto 
[  35.719  56.422    34.921  55.256    34.497  54.494    34.228  53.918   
 34.039  53.439    33.892  53.001    33.793  52.651    33.714  52.315   
 33.651  51.992    33.612  51.733    33.581  51.478    33.560  51.226   
 33.547  50.992    33.542  50.797    33.542  50.604    33.548  50.413   
 33.557  50.225    33.572  50.039    33.590  49.855    33.613  49.674   
 33.640  49.495    33.671  49.318    33.705  49.143    33.743  48.970   
 33.785  48.799    33.830  48.631    33.878  48.464    33.929  48.300   
 33.983  48.137    34.039  47.977    34.099  47.818    34.161  47.661   
 34.225  47.507    34.292  47.354    34.361  47.203    34.432  47.054   
 34.506  46.907    34.581  46.761    34.658  46.617    34.736  46.476   
 34.817  46.335    34.899  46.197    34.983  46.060    35.067  45.925   
 35.154  45.791    35.241  45.660    35.330  45.529    35.420  45.401   
 35.511  45.274    35.603  45.148    35.696  45.024    35.790  44.902   
 35.885  44.781    35.980  44.661    36.076  44.543    36.173  44.426   
 36.270  44.311    36.368  44.198    36.467  44.085    36.566  43.974   
 36.665  43.865    36.764  43.756    36.864  43.649    36.965  43.544   
 37.065  43.439    37.166  43.336    37.267  43.234    37.368  43.134   
 37.469  43.034    37.570  42.936    37.671  42.839    37.772  42.743   
 37.873  42.649    37.974  42.555    38.075  42.463    38.176  42.372   
 38.277  42.282    38.377  42.193    38.477  42.105    38.577  42.018   
 38.677  41.932    38.777  41.847    38.876  41.764    38.975  41.681   
 39.074  41.599    39.172  41.519    39.270  41.439    39.368  41.361   
 39.465  41.283    39.562  41.206    39.659  41.130    39.755  41.055   
 39.850  40.981    39.946  40.908    40.040  40.836    40.135  40.765   
 40.228  40.694    40.322  40.625    40.414  40.556    40.507  40.488   
 40.598  40.421    40.689  40.355    40.780  40.289    40.870  40.225   
 40.960  40.161    41.049  40.098    41.137  40.035    41.225  39.974   
 41.312  39.913    41.399  39.853    41.485  39.794    41.571  39.735   
 41.655  39.677    41.740  39.620    41.823  39.564    41.907  39.508   
 41.989  39.453    42.071  39.398    42.152  39.345    42.233  39.292   
 42.313  39.239    42.392  39.187    42.471  39.136    42.549  39.086   
 42.627  39.036    42.704  38.986    42.780  38.938    42.856  38.890   
 42.931  38.842    43.005  38.795    43.079  38.749    43.152  38.703   
 43.225  38.658    43.297  38.613    43.368  38.569    43.439  38.525   
 43.509  38.482    43.579  38.440    43.648  38.398    43.716  38.356   
 43.784  38.315    43.851  38.275    43.917  38.235    43.983  38.195   
 44.048  38.156    44.113  38.117    44.177  38.079    44.241  38.042   
 44.304  38.004    44.366  37.968    44.428  37.931    44.489  37.896   
 44.549  37.860    44.609  37.825    44.669  37.791    44.728  37.757   
 44.786  37.723    44.844  37.690    44.901  37.657    44.957  37.624   
 45.013  37.592    45.069  37.561    45.124  37.529    45.178  37.498   
 45.232  37.468    45.285  37.438    45.338  37.408    45.390  37.379   
 45.442  37.349    45.493  37.321    45.544  37.292    45.594  37.264   
 45.644  37.237    45.693  37.209    45.742  37.182    45.790  37.156   
 45.838  37.129    45.885  37.103    45.932  37.078    45.978  37.052   
 46.024  37.027    46.069  37.002    46.114  36.978    46.158  36.954   
 46.202  36.930    46.245  36.906    46.288  36.883    46.331  36.860   
 46.373  36.837    46.414  36.815   ] scurve 
[  46.414  36.815    46.455  36.793    46.496  36.771    46.536  36.749   
 46.576  36.728    46.616  36.707    46.655  36.686    46.693  36.665   
 46.731  36.645    46.769  36.625    46.806  36.605    46.843  36.585   
 46.880  36.566    46.916  36.547    46.952  36.528    46.987  36.509   
 47.022  36.491    47.057  36.472    47.091  36.454    47.125  36.437   
 47.158  36.419    47.191  36.402    47.224  36.385    47.256  36.368   
 47.288  36.351    47.320  36.335    47.351  36.318    47.382  36.302   
 47.413  36.286    47.443  36.270    47.473  36.255    47.503  36.240   
 47.532  36.224    47.561  36.209    47.589  36.195    47.618  36.180   
 47.646  36.166    47.673  36.151    47.701  36.137    47.728  36.123   
 47.755  36.110    47.781  36.096    47.807  36.083    47.833  36.070   
 47.858  36.056    47.884  36.044    47.909  36.031    47.933  36.018   
 47.958  36.006    47.982  35.993    48.006  35.981    48.029  35.969   
 48.053  35.957    48.076  35.946    48.099  35.934    48.121  35.923   
 48.143  35.912    48.165  35.900    48.187  35.889    48.209  35.879   
 48.230  35.868    48.251  35.857    48.272  35.847    48.292  35.836   
 48.312  35.826    48.333  35.816    48.352  35.806    48.372  35.796   
 48.391  35.786    48.410  35.777    48.429  35.767    48.448  35.758   
 48.467  35.749    48.485  35.740    48.503  35.731    48.521  35.722   
 48.538  35.713    48.556  35.704    48.573  35.695    48.590  35.687   
 48.607  35.679    48.624  35.670    48.640  35.662    48.656  35.654   
 48.672  35.646    48.688  35.638    48.704  35.630    48.720  35.623   
 48.735  35.615    48.750  35.608    48.765  35.600    48.780  35.593   
 48.794  35.586    48.809  35.578    48.823  35.571    48.837  35.564   
 48.851  35.557    48.865  35.551    48.879  35.544    48.892  35.537   
 48.905  35.531    50.000  35.000  ] scurve 
 35.719  56.422 moveto 
[  35.719  56.422    37.134  58.050    39.054  59.802    41.139  61.384   
 42.863  62.533    44.430  63.488    45.960  64.356    47.355  65.102   
 48.795  65.834   ] scurve 
 35.719  13.578 moveto 
[  35.719  13.578    35.252  14.224    35.118  14.428    34.990  14.629   
 34.869  14.828    34.754  15.026    34.645  15.221    34.543  15.415   
 34.446  15.606    34.355  15.796    34.270  15.984    34.190  16.170   
 34.116  16.354    34.047  16.537    33.982  16.718    33.923  16.897   
 33.869  17.074    33.819  17.249    33.774  17.423    33.733  17.595   
 33.696  17.765    33.664  17.934    33.635  18.101    33.611  18.266   
 33.590  18.430    33.574  18.592    33.560  18.753    33.551  18.911   
 33.544  19.069    33.541  19.225    33.542  19.379    33.545  19.532   
 33.551  19.683    33.561  19.833    33.573  19.982    33.588  20.129   
 33.606  20.274    33.626  20.418    33.649  20.561    33.674  20.702   
 33.702  20.842    33.732  20.981    33.764  21.118    33.798  21.254   
 33.835  21.388    33.873  21.521    33.914  21.653    33.956  21.784   
 34.000  21.913    34.046  22.041    34.094  22.168    34.143  22.293   
 34.196  22.423    34.262  22.577    34.330  22.729    34.400  22.879   
 34.479  23.040    34.573  23.222    34.669  23.402    34.777  23.595   
 34.905  23.812    35.039  24.029    35.208  24.290    35.402  24.572   
 35.636  24.894    35.943  25.291    36.374  25.807    37.075  26.570   
 37.828  27.307    38.578  27.980    39.315  28.594    40.032  29.154   
 40.722  29.666    41.383  30.132    42.011  30.558    42.605  30.946   
 43.165  31.301    43.690  31.624    44.181  31.919    44.640  32.188   
 45.067  32.434    45.464  32.658    45.832  32.863    46.172  33.050   
 46.487  33.220    46.777  33.376    47.045  33.518    47.292  33.647   
 47.519  33.766    47.728  33.873    47.920  33.972    48.097  34.062   
 48.259  34.144    48.407  34.219    48.543  34.287    48.668  34.349   
 48.782  34.406    48.887  34.458    48.983  34.505    49.070  34.549   
 49.151  34.588    49.224  34.624    49.291  34.657    49.353  34.687   
 49.409  34.714    49.460  34.739    49.507  34.762    49.550  34.783   
 49.589  34.802    49.625  34.819    49.657  34.835    49.687  34.849   
 49.714  34.862    49.739  34.874    49.762  34.885    49.783  34.895   
 49.802  34.905    49.819  34.913    49.835  34.921    49.849  34.927   
 49.862  34.934    49.874  34.940    49.885  34.945    49.895  34.950   
 49.905  34.954    49.913  34.958    50.000  35.000  ] scurve 
 35.719  13.578 moveto 
[  35.719  13.578    36.253  12.913    36.441  12.694    36.638  12.474   
 36.843  12.252    37.057  12.027    37.280  11.800    37.513  11.571   
 37.754  11.340    38.005  11.106    38.266  10.870    38.537  10.631   
 38.819  10.391    39.111  10.148    39.414   9.902    39.728   9.654   
 40.054   9.404    40.391   9.151    40.740   8.896    41.035   8.685   
 41.331   8.477    41.636   8.268    41.949   8.057    42.270   7.844   
 42.600   7.630    42.939   7.414    43.287   7.196    43.644   6.977   
 44.010   6.756    44.385   6.533    44.771   6.308    45.166   6.081   
 45.571   5.853    45.987   5.623    46.413   5.391    46.849   5.157   
 47.296   4.922    47.755   4.684    48.224   4.445   ] scurve 
 28.578  42.141 moveto 
[  28.578  42.141    29.404  41.815    29.725  41.690    30.041  41.568   
 30.354  41.448    30.663  41.330    30.968  41.214    31.269  41.101   
 31.567  40.989    31.860  40.880    32.150  40.772    32.436  40.667   
 32.719  40.563    32.997  40.461    33.272  40.362    33.543  40.264   
 33.810  40.167    34.074  40.073    34.334  39.980    34.590  39.889   
 34.843  39.800    35.092  39.712    35.338  39.626    35.580  39.541   
 35.818  39.458    36.054  39.377    36.285  39.297    36.514  39.218   
 36.739  39.141    36.960  39.065    37.179  38.991    37.394  38.918   
 37.605  38.847    37.814  38.776    38.019  38.707    38.222  38.639   
 38.421  38.573    38.617  38.508    38.810  38.443    39.000  38.380   
 39.188  38.319    39.372  38.258    39.553  38.198    39.732  38.140   
 39.907  38.083    40.080  38.026    40.250  37.971    40.418  37.917   
 40.583  37.863    40.745  37.811    40.904  37.760    41.061  37.709   
 41.216  37.660    41.368  37.611    41.517  37.563    41.664  37.516   
 41.809  37.470    41.951  37.425    42.091  37.381    42.229  37.337   
 42.364  37.295    42.497  37.253    42.628  37.211    42.757  37.171   
 42.883  37.131    43.008  37.092    43.130  37.054    43.251  37.017   
 43.369  36.980    43.486  36.944    43.600  36.908    43.712  36.873   
 43.823  36.839    43.932  36.805    44.039  36.772    44.144  36.740   
 44.247  36.708    44.349  36.677    44.448  36.646    44.547  36.616   
 44.643  36.587    44.738  36.558    44.831  36.529    44.923  36.501   
 45.013  36.474    45.101  36.447    45.188  36.420    45.274  36.394   
 45.357  36.369    45.440  36.344    45.521  36.319    45.601  36.295   
 45.679  36.271    45.756  36.248    45.832  36.225    45.906  36.203   
 45.979  36.181    46.051  36.159    46.121  36.138    46.191  36.117   
 46.259  36.097    46.326  36.077    46.391  36.057    46.456  36.038   
 46.519  36.019    46.582  36.000    46.643  35.982    46.703  35.964   
 46.762  35.946    46.820  35.929    46.877  35.912    46.933  35.895   
 46.988  35.879    47.042  35.863    47.095  35.847    47.148  35.832   
 47.199  35.817    47.249  35.802    47.299  35.787    47.347  35.773   
 47.395  35.758    47.442  35.745    47.488  35.731    47.533  35.718   
 47.578  35.704    47.621  35.692    47.664  35.679    47.706  35.667   
 47.747  35.654    47.788  35.642    47.828  35.631    47.867  35.619   
 47.906  35.608    47.943  35.597    47.981  35.586    48.017  35.575   
 48.053  35.565    48.088  35.554    48.123  35.544    48.157  35.534   
 48.190  35.524    48.223  35.515    48.255  35.505    48.286  35.496   
 48.317  35.487    48.348  35.478    48.378  35.469    48.407  35.461   
 48.436  35.452    48.464  35.444    48.492  35.436    48.519  35.428   
 48.546  35.420    48.572  35.413    48.598  35.405    48.624  35.398   
 48.649  35.390    48.673  35.383    48.697  35.376    50.000  35.000  ] scurve 
 28.578  42.141 moveto 
[  28.578  42.141    27.725  42.480    27.381  42.619    27.034  42.761   
 26.682  42.905    26.327  43.052    25.969  43.201    25.607  43.354   
 25.241  43.509    24.872  43.667    24.499  43.828    24.123  43.992   
 23.744  44.159    23.361  44.329    22.975  44.503    22.586  44.679   
 22.195  44.859    21.800  45.042    21.403  45.229    21.003  45.419   
 20.681  45.574    20.358  45.731    20.033  45.890    19.707  46.052   
 19.380  46.216    19.051  46.382    18.721  46.551    18.391  46.723   
 18.059  46.897    17.727  47.073    17.394  47.252    17.060  47.434   
 16.726  47.619    16.392  47.806    16.057  47.996    15.722  48.189   
 15.388  48.385    15.054  48.584    14.720  48.785    14.387  48.990   
 14.054  49.197    13.722  49.408    13.392  49.622    13.063  49.839   
 12.783  50.027    12.522  50.205    12.262  50.386    12.004  50.568   
 11.747  50.753    11.491  50.940    11.238  51.129    10.986  51.320   
 10.736  51.513    10.488  51.709    10.242  51.907     9.999  52.108   
  9.759  52.311     9.521  52.516     9.286  52.724     9.054  52.934   
  8.826  53.147     8.601  53.362     8.379  53.580     8.161  53.800   
  7.948  54.023     7.738  54.249     7.533  54.477     7.333  54.708   
  7.137  54.942     6.947  55.178     6.762  55.418     6.605  55.629   
  6.465  55.824     6.328  56.022     6.196  56.221     6.068  56.422   
  5.944  56.625     5.825  56.830     5.710  57.037     5.600  57.246   
  5.494  57.457     5.394  57.670     5.299  57.885     5.209  58.101   
  5.125  58.320     5.046  58.541     4.974  58.765     4.907  58.990   
  4.846  59.217     4.792  59.447   ] scurve 
 28.578  27.859 moveto 
[  28.578  27.859    29.607  28.262    30.377  28.558    31.124  28.841   
 31.848  29.112    32.550  29.370    33.229  29.618    33.886  29.854   
 34.521  30.080    35.134  30.297    35.726  30.503    36.298  30.701   
 36.849  30.890    37.380  31.070    37.892  31.243    38.385  31.408   
 38.860  31.566    39.317  31.717    39.757  31.861    40.180  31.999   
 40.587  32.131    40.978  32.257    41.354  32.377    41.715  32.493   
 42.062  32.603    42.395  32.708    42.715  32.809    43.022  32.905   
 43.317  32.997    43.600  33.085    43.871  33.169    44.131  33.250   
 44.381  33.326    44.620  33.400    44.850  33.470    45.070  33.537   
 45.281  33.602    45.483  33.663    45.677  33.722    45.862  33.778   
 46.040  33.832    46.211  33.883    46.374  33.932    46.530  33.979   
 46.680  34.024    46.823  34.067    46.961  34.108    47.092  34.147   
 47.218  34.184    47.339  34.220    47.454  34.254    47.565  34.287   
 47.670  34.319    47.771  34.348    47.868  34.377    47.961  34.404   
 48.050  34.431    48.135  34.456    48.216  34.480    48.294  34.502   
 48.368  34.524    48.439  34.545    48.507  34.565    48.572  34.584   
 48.635  34.602    48.694  34.620    48.751  34.637    48.806  34.653   
 48.858  34.668    48.908  34.682    48.956  34.696    49.002  34.710   
 49.045  34.722    49.087  34.735    49.127  34.746    49.165  34.757   
 49.202  34.768    49.237  34.778    49.270  34.788    49.302  34.797   
 49.333  34.806    49.362  34.815    49.390  34.823    49.417  34.831   
 49.442  34.838    49.467  34.845    49.490  34.852    50.000  35.000  ] scurve 
 28.578  27.859 moveto 
[  28.578  27.859    27.512  27.431    26.662  27.083    25.789  26.718   
 24.896  26.336    23.982  25.937    23.048  25.520    22.097  25.083   
 21.129  24.627    20.146  24.149    19.150  23.649    18.145  23.126   
 17.132  22.580    16.115  22.008    15.097  21.410    14.084  20.784   
 13.081  20.129    12.092  19.445    11.124  18.729    10.185  17.980   
  9.282  17.197     8.426  16.377     7.625  15.520     6.893  14.624   
  6.241  13.686     5.749  12.833     5.446  12.198     5.234  11.666   
  5.086  11.222     4.974  10.821     4.893  10.462     4.833  10.131   
] scurve 
 21.437  56.422 moveto 
[  21.437  56.422    21.407  55.397    21.418  55.192    21.435  54.989   
 21.457  54.787    21.484  54.588    21.516  54.390    21.553  54.195   
 21.595  54.002    21.641  53.810    21.691  53.621    21.746  53.433   
 21.804  53.247    21.867  53.064    21.933  52.882    22.003  52.701   
 22.077  52.523    22.154  52.347    22.234  52.172    22.318  51.999   
 22.405  51.828    22.494  51.658    22.587  51.490    22.682  51.324   
 22.780  51.160    22.881  50.997    22.983  50.836    23.089  50.676   
 23.196  50.518    23.306  50.362    23.418  50.207    23.532  50.054   
 23.648  49.902    23.766  49.752    23.885  49.603    24.006  49.456   
 24.129  49.311    24.254  49.166    24.380  49.024    24.507  48.882   
 24.635  48.742    24.765  48.604    24.896  48.467    25.029  48.331   
 25.162  48.197    25.296  48.064    25.432  47.932    25.568  47.802   
 25.705  47.673    25.843  47.545    25.981  47.419    26.121  47.294   
 26.261  47.170    26.401  47.047    26.542  46.926    26.684  46.806   
 26.826  46.687    26.969  46.569    27.112  46.452    27.255  46.337   
 27.399  46.223    27.543  46.110    27.687  45.998    27.831  45.887   
 27.975  45.777    28.120  45.669    28.265  45.561    28.409  45.455   
 28.554  45.349    28.699  45.245    28.844  45.142    28.988  45.040   
 29.133  44.939    29.277  44.838    29.422  44.739    29.566  44.641   
 29.710  44.544    29.854  44.448    29.997  44.353    30.141  44.258   
 30.284  44.165    30.426  44.073    30.569  43.981    30.711  43.891   
 30.852  43.801    30.994  43.713    31.135  43.625    31.275  43.538   
 31.415  43.452    31.555  43.367    31.694  43.282    31.833  43.199   
 31.971  43.116    32.109  43.034    32.246  42.953    32.383  42.873   
 32.519  42.794    32.655  42.715    32.790  42.638    32.925  42.561   
 33.059  42.485    33.192  42.409    33.325  42.334    33.457  42.261   
 33.588  42.187    33.719  42.115    33.850  42.043    33.979  41.972   
 34.108  41.902    34.237  41.832    34.364  41.764    34.492  41.695   
 34.618  41.628    34.744  41.561    34.869  41.495    34.993  41.430   
 35.117  41.365    35.240  41.301    35.362  41.237    35.483  41.174   
 35.604  41.112    35.724  41.051    35.844  40.990    35.963  40.929   
 36.081  40.869    36.198  40.810    36.314  40.752    36.430  40.694   
 36.545  40.636    36.660  40.580    36.773  40.523    36.886  40.468   
 36.999  40.413    37.110  40.358    37.221  40.304    37.331  40.251   
 37.440  40.198    37.549  40.145    37.656  40.094    37.763  40.042   
 37.870  39.991    37.975  39.941    38.080  39.891    38.184  39.842   
 38.288  39.793    38.391  39.745    38.492  39.697    38.594  39.650   
 38.694  39.603    38.794  39.556    38.893  39.511    38.991  39.465   
 39.089  39.420    39.186  39.376    39.282  39.331    39.378  39.288   
 39.472  39.245    39.566  39.202    39.660  39.159    39.752  39.118   
 39.844  39.076    39.936  39.035    40.026  38.994    40.116  38.954   
 40.205  38.914    40.294  38.875    40.381  38.836    40.468  38.797   
 40.555  38.759    40.641  38.721    40.726  38.683    40.810  38.646   
 40.894  38.610    40.977  38.573    41.059  38.537    41.141  38.502   
 41.222  38.466    41.302  38.431    41.382  38.397    41.461  38.363   
 41.540  38.329    41.618  38.295    41.695  38.262    41.771  38.229   
 41.847  38.196    41.923  38.164    41.997  38.132    42.072  38.101   
 42.145  38.070    42.218  38.039   ] scurve 
[  42.218  38.039    42.290  38.008    42.362  37.978    42.433  37.948   
 42.503  37.918    42.573  37.889    42.642  37.859    42.711  37.831   
 42.779  37.802    42.847  37.774    42.913  37.746    42.980  37.718   
 43.046  37.691    43.111  37.664    43.176  37.637    43.240  37.610   
 43.303  37.584    43.366  37.558    43.429  37.532    43.491  37.507   
 43.552  37.481    43.613  37.456    43.673  37.432    43.733  37.407   
 43.792  37.383    43.851  37.359    43.909  37.335    43.967  37.312   
 44.024  37.288    44.081  37.265    44.137  37.242    44.193  37.220   
 44.248  37.197    44.303  37.175    44.357  37.153    44.411  37.132   
 44.464  37.110    44.517  37.089    44.569  37.068    44.621  37.047   
 44.672  37.026    44.723  37.006    44.774  36.986    44.824  36.966   
 44.873  36.946    44.922  36.926    44.971  36.907    45.019  36.888   
 45.067  36.869    45.115  36.850    45.161  36.831    45.208  36.813   
 45.254  36.794    45.300  36.776    45.345  36.758    45.390  36.741   
 45.434  36.723    45.478  36.706    45.522  36.689    45.565  36.672   
 45.608  36.655    45.650  36.638    45.692  36.622    45.734  36.605   
 45.775  36.589    45.816  36.573    45.857  36.557    45.897  36.541   
 45.936  36.526    45.976  36.511    46.015  36.495    46.054  36.480   
 46.092  36.465    46.130  36.451    46.167  36.436    46.205  36.421   
 46.241  36.407    46.278  36.393    46.314  36.379    46.350  36.365   
 46.386  36.351    46.421  36.338    46.456  36.324    46.490  36.311   
 46.524  36.298    46.558  36.285    46.592  36.272    46.625  36.259   
 46.658  36.246    46.691  36.234    46.723  36.221    46.755  36.209   
 46.787  36.197    46.818  36.185    46.849  36.173    46.880  36.161   
 46.911  36.149    46.941  36.138    46.971  36.126    47.000  36.115   
 47.030  36.103    50.000  35.000  ] scurve 
 21.437  56.422 moveto 
[  21.437  56.422    21.615  57.503    21.709  57.862    21.822  58.226   
 21.955  58.596    22.109  58.972    22.284  59.355    22.483  59.743   
 22.705  60.137    22.952  60.538    23.225  60.945    23.547  61.387   
 24.024  61.987    24.780  62.834    25.660  63.707    26.675  64.608   
 27.835  65.537   ] scurve 
 21.437  13.578 moveto 
[  21.437  13.578    21.407  14.603    21.422  14.859    21.445  15.113   
 21.477  15.363    21.516  15.610    21.563  15.854    21.617  16.094   
 21.678  16.332    21.746  16.567    21.820  16.799    21.900  17.028   
 21.986  17.254    22.077  17.477    22.174  17.697    22.276  17.915   
 22.383  18.130    22.495  18.342    22.611  18.551    22.754  18.796   
 22.911  19.050    23.108  19.351    23.341  19.685    23.645  20.092   
 24.080  20.629    24.796  21.426    25.595  22.221    26.424  22.969   
 27.273  23.674    28.132  24.337    28.995  24.961    29.855  25.549   
 30.707  26.102    31.546  26.623    32.369  27.113    33.173  27.574   
 33.956  28.009    34.716  28.418    35.452  28.803    36.163  29.166   
 36.848  29.507    37.508  29.828    38.141  30.131    38.749  30.416   
 39.331  30.684    39.887  30.936    40.419  31.174    40.926  31.398   
 41.410  31.609    41.871  31.807    42.309  31.994    42.727  32.169   
 43.123  32.335    43.500  32.491    43.857  32.638    44.197  32.776   
 44.518  32.906    44.823  33.028    45.112  33.144    45.386  33.252   
 45.645  33.354    45.890  33.451    46.122  33.541    46.341  33.627   
 46.548  33.707    46.744  33.782    46.930  33.854    47.105  33.921   
 47.270  33.984    47.426  34.043    47.573  34.099    47.712  34.152   
 47.844  34.201    47.968  34.248    48.085  34.292    48.195  34.333   
 48.299  34.372    48.397  34.409    48.490  34.444    48.577  34.476   
 48.659  34.507    48.737  34.536    48.810  34.563    48.879  34.588   
 48.944  34.612    49.005  34.635    49.063  34.656    49.118  34.677   
 49.169  34.695    49.217  34.713    49.263  34.730    50.000  35.000  ] scurve 
 21.437  13.578 moveto 
[  21.437  13.578    21.612  12.504    21.670  12.275    21.735  12.044   
 21.809  11.810    21.890  11.575    21.980  11.336    22.078  11.096   
 22.186  10.852    22.302  10.607    22.428  10.359    22.563  10.108   
 22.708   9.855    22.838   9.641    22.968   9.435    23.105   9.227   
 23.249   9.017    23.401   8.806    23.559   8.593    23.725   8.378   
 23.899   8.162    24.080   7.944    24.269   7.724    24.466   7.502   
 24.672   7.278    24.885   7.053    25.108   6.826    25.339   6.597   
 25.579   6.366    25.829   6.133    26.087   5.898    26.355   5.662   
 26.633   5.423    26.921   5.183    27.219   4.941    27.527   4.696   
 27.846   4.450   ] scurve 
  7.156  27.859 moveto 
[   7.156  27.859     8.527  28.106     9.676  28.312    10.796  28.511   
 11.888  28.705    12.952  28.893    13.989  29.076    14.999  29.252   
 15.983  29.424    16.941  29.591    17.873  29.752    18.782  29.909   
 19.666  30.061    20.526  30.208    21.364  30.351    22.179  30.490   
 22.973  30.625    23.744  30.755    24.495  30.882    25.226  31.005   
 25.936  31.124    26.628  31.240    27.300  31.352    27.953  31.461   
 28.589  31.567    29.207  31.669    29.808  31.769    30.392  31.865   
 30.959  31.959    31.511  32.050    32.048  32.138    32.569  32.223   
 33.076  32.306    33.568  32.386    34.047  32.464    34.512  32.540   
 34.964  32.614    35.403  32.685    35.829  32.754    36.243  32.821   
 36.646  32.886    37.037  32.949    37.417  33.010    37.785  33.070   
 38.144  33.127    38.492  33.183    38.830  33.237    39.158  33.290   
 39.477  33.341    39.786  33.391    40.087  33.439    40.379  33.485   
 40.662  33.531    40.938  33.574    41.205  33.617    41.464  33.658   
 41.716  33.698    41.961  33.737    42.198  33.775    42.429  33.811   
 42.653  33.847    42.870  33.881    43.081  33.915    43.286  33.947   
 43.485  33.979    43.677  34.009    43.865  34.039    44.047  34.067   
 44.223  34.095    44.394  34.122    44.561  34.148    44.722  34.174   
 44.879  34.198    45.031  34.222    45.178  34.246    45.321  34.268   
 45.460  34.290    45.595  34.311    45.726  34.332    45.853  34.352   
 45.976  34.371    46.096  34.390    46.212  34.408    46.325  34.426   
 46.434  34.443    46.540  34.459    46.643  34.476    46.743  34.491   
 46.840  34.506    46.934  34.521    47.025  34.535    47.114  34.549   
 47.200  34.563    47.283  34.576    47.364  34.588    47.443  34.601   
 47.519  34.613    47.593  34.624    47.665  34.635    47.734  34.646   
 47.802  34.657    47.867  34.667    47.931  34.677    47.993  34.687   
 48.053  34.696    48.111  34.705    48.167  34.714    48.222  34.722   
 48.275  34.731    48.326  34.739    48.376  34.747    48.425  34.754   
 48.472  34.761    48.517  34.769    48.562  34.776    48.605  34.782   
 48.646  34.789    50.000  35.000  ] scurve 
  7.156  27.859 moveto 
[   7.156  27.859     5.745  27.602     4.497  27.374   ] scurve 
  7.156  42.141 moveto 
[   7.156  42.141     8.275  41.938     9.018  41.804     9.748  41.673   
 10.467  41.544    11.174  41.418    11.870  41.294    12.554  41.172   
 13.227  41.053    13.888  40.937    14.539  40.822    15.179  40.710   
 15.808  40.600    16.427  40.492    17.035  40.386    17.633  40.282   
 18.221  40.180    18.799  40.080    19.368  39.982    19.926  39.886   
 20.475  39.791    21.014  39.699    21.545  39.608    22.066  39.519   
 22.578  39.432    23.081  39.347    23.575  39.263    24.061  39.181   
 24.538  39.100    25.007  39.021    25.468  38.943    25.921  38.867   
 26.366  38.793    26.803  38.719    27.232  38.648    27.653  38.577   
 28.067  38.508    28.474  38.441    28.874  38.374    29.266  38.309   
 29.651  38.245    30.030  38.183    30.402  38.121    30.767  38.061   
 31.125  38.002    31.477  37.944    31.823  37.887    32.162  37.831   
 32.496  37.777    32.823  37.723    33.144  37.671    33.460  37.619   
 33.770  37.569    34.074  37.519    34.372  37.471    34.666  37.423   
 34.954  37.376    35.236  37.330    35.514  37.285    35.786  37.241   
 36.053  37.198    36.316  37.156    36.573  37.114    36.826  37.073   
 37.074  37.033    37.318  36.994    37.557  36.955    37.792  36.918   
 38.022  36.881    38.248  36.844    38.470  36.809    38.688  36.774   
 38.902  36.740    39.112  36.706    39.318  36.673    39.520  36.641   
 39.718  36.609    39.913  36.578    40.104  36.548    40.292  36.518   
 40.476  36.489    40.656  36.460    40.834  36.432    41.007  36.404   
 41.178  36.377    41.346  36.350    41.510  36.324    41.671  36.299   
 41.829  36.274    41.985  36.249    42.137  36.225    42.287  36.201   
 42.433  36.178    42.577  36.156    42.718  36.133    42.857  36.111   
 42.993  36.090    43.126  36.069    43.257  36.048    43.386  36.028   
 43.512  36.008    43.635  35.989    43.757  35.970    43.876  35.951   
 43.993  35.933    44.107  35.915    44.220  35.897    44.330  35.880   
 44.438  35.863    44.544  35.846    44.648  35.830    44.750  35.814   
 44.851  35.798    44.949  35.783    45.046  35.767    45.140  35.753   
 45.233  35.738    45.324  35.724    45.414  35.710    45.501  35.696   
 45.587  35.683    45.672  35.670    45.754  35.657    45.836  35.644   
 45.915  35.632    45.993  35.619    46.070  35.607    46.145  35.596   
 46.219  35.584    46.291  35.573    46.362  35.562    46.432  35.551   
 46.500  35.540    46.567  35.530    46.633  35.520    46.698  35.510   
 46.761  35.500    46.823  35.490    46.884  35.481    46.943  35.472   
 47.002  35.462    47.059  35.454    47.116  35.445    47.171  35.436   
 47.225  35.428    47.279  35.420    47.331  35.411    47.382  35.403   
 47.432  35.396    47.481  35.388    47.530  35.381    47.577  35.373   
 47.624  35.366    47.669  35.359    47.714  35.352    47.758  35.345   
 47.801  35.339    47.843  35.332    47.884  35.326    47.925  35.319   
 47.965  35.313    48.004  35.307    48.042  35.301    48.080  35.295   
 48.117  35.290    50.000  35.000  ] scurve 
  7.156  42.141 moveto 
[   7.156  42.141     6.008  42.348     5.216  42.492     4.411  42.639   
] scurve 
 50.000  50.000 moveto 
  0.700   0.000 indent 
 65.000  35.000 moveto 
  0.700  90.000 indent 
grestore}
\LabelTeX   98.000  31.200 $x$\ELTX
\LabelTeX   46.500  68.000 $y$\ELTX
\LabelTeX   44.300  56.000 $-\lambda$\ELTX
\LabelTeX   66.000  31.500 $1$\ELTX
\LabelTeX   47.000  49.000 $1$\ELTX
\EndFig
\vskip4mm

\subsection 1.4. \'Equations lin\'eaires du premier ordre|

\noindent Ce sont les \'equations de la forme
$$
y'=a(x)y+b(x)\leqno{\rm (E)}
$$
o\`u $a,b: I\rightarrow\bR$ (ou $\bC$) sont des fonctions continues.

Supposons qu'on connaisse une solution particuli\`ere $y_{(1)}$ de
l'\'equation (E). Alors on obtient par soustraction
$y'-y'_{(1)}=a(x)(y-y_{(1)})$, c'est-\`a-dire que $z=y-y_{(1)}$
v\'erifie l'\'equation lin\'eaire \og$\,$sans second membre$\,$\fg
$$
z'=a(x)z.\leqno({\rm E}_0)
$$
Inversement, si $z$ est solution de ({\rm E}$_0$), alors $y=y_{(1)}+z$ est solution de
(E).

\bigskip
\majdeb
\state Th\'eor\`eme 1 -- |  {\it La solution g\'en\'erale de {\rm (E)} s'\'ecrit
$$
y=y_{(1)}+z
$$
o\`u $y_{(1)}$ est une solution particuli\`ere de {\rm (E)} et o\`u  $z$ est la solution
g\'en\'erale de {\rm (E}$_0)$.}
\par\majfin\par

\bigskip
\noindent{\it {\rm a)} Solutions de {\rm (E}$_0)$}
\medskip

Comme $f(x,z)=a(x)z$ est continue et de d\'eriv\'ee partielle $\dfrac{\partial
f}{\partial z}\ (x,z)=a(x)$ continue, on sait que le probl\`eme de Cauchy admet une
solution unique en tout point $(x_0,z_0)\in I\times\bR$. Or $z(x)\equiv 0$ est clairement
solution de (E$_0$). D'apr\`es l'unicit\'e, aucune autre solution ne peut s'annuler en un
quelconque point $x_0\in I$. Si $z\not=0$, on peut donc \'ecrire 
$$
\eqalign{
\frac{z'}{z}&=a(x),\cr
\ln\,|z|&=A(x)+C,\quad C\in\bR,\cr
}
$$
o\`u $A$ est une primitive de $a$ sur $I$. On en d\'eduit
$$
\eqalign{
|z(x)|&=e^Ce^{A(x)},\cr
z(x)&=\varepsilon(x)e^Ce^{A(x)}\quad{\rm avec}\quad \varepsilon(x)=\pm 1.\cr
}
$$
Comme $z$ est continue et ne s'annule pas, le signe de $z$ ne change pas, d'o\`u
$$
z(x)=\lambda e^{A(x)}
$$
avec $\lambda=\pm e^C$. Inversement, toute fonction
$$
z(x)=\lambda e^{A(x)},\quad \lambda\in\bR
$$
et visiblement solution de ({\rm E}$_0$). On peut donc \'enoncer~:

\bigskip
\majdeb
\state Th\'eor\`eme 2 -- |  {\it Les solutions maximales de {\rm (E}$_0)$~:
$z'=a(x)z$ forment un espace vectoriel de dimension 1, ayant pour base $x\mapsto
e^{A(x)}$.}  
\par\majfin\par

\bigskip
\noindent b) {\it Recherche d'une solution particuli\`ere $y_{(1)}$ de {\rm (E)}.}
\medskip

Si aucune solution \'evidente n'appara\^{\i}t, on peut utiliser la m\'ethode dite de {\it
variation des constantes}, c'est-\`a-dire que l'on cherche $y_{(1)}$ sous la forme
$$
y_{(1)}(x)=\lambda(x)e^{A(x)},
$$
o\`u $\lambda$ est diff\'erentiable. Il vient
$$
\eqalign{
y'_{(1)}(x)&=\lambda(x)a(x)e^{A(x)}+\lambda'(x)e^{A(x)}\cr
&=a(x)y_{(1)}(x)+\lambda'(x)e^{A(x)}.\cr
}
$$
$y_{(1)}$ est donc solution de (E) si on prend
$$
\eqalign{
&\lambda'(x)e^{A(x)}=b(x),\cr
&\lambda'(x)=b(x)e^{-A(x)},\cr
&\lambda(x)=\int^x_{x_0}b(t)e^{-A(t)},\quad x_0\in I.\cr
}
$$
On obtient ainsi la solution particuli\`ere
$$
y_{(1)}(x)=e^{A(x)}\int^x_{x_0}b(t)e^{-A(t)}dt
$$
telle que $y_{(1)}(x_0)=0$. La solution g\'en\'erale est donn\'ee d'apr\`es le
th\'eor\`eme 1 par

\bigskip
\majdeb
\hfil$\displaystyle
y(x)=e^{A(x)}\Big(\lambda+\int^x_{x_0}b(t)e^{-A(t)}dt\Big)$.\hfill \par\majfin\par

\bigskip
La solution du probl\`eme de Cauchy $y(x_0)=y_0$ est obtenue pour
$\lambda=e^{-A(x_0)}y_0$.

%
\bigskip
\majedeb
\state Exercice -- |  {\it Propri\'et\'es g\'eom\'etriques li\'ees 
aux \'equations lin\'eaires $($cf.\ sch\'ema$)$.
\medskip
{\parindent1.6pc
\item{\rm (a)} Si $y_{(1)}$, $y_{(2)}$, $y_{(3)}$ sont trois solutions
d'une \'equation lin\'eaire, montrer que la fonction $y_{(3)}-y_{(2)}$ 
est proportionnelle \`a $y_{(2)}-y_{(1)}$.
\medskip
\item{\rm (b)} Montrer qu'une \'equation $y'=f(x,y)$ est lin\'eaire si et
seulement si le champ des tangentes a la propri\'et\'e suivante~: pour tout
$x_0$ fix\'e, les tangentes aux diff\'erents points $(x_0,y)$ sont 
concourantes ou toutes parall\`eles.\par}
}
\par\majefin\par

\InsertFig 16.000 82.000 {
1 mm unit
  5.000  25.000 moveto 
100.000   0.000 vector 
 58.000  15.000 moveto 
 55.000  90.000 vector 
  5.000  15.000 moveto 
 90.000  50.000 rect  clip  newpath 
  0.340 setlinewidth
 12.258  65.316 moveto 
[  12.258  65.316    12.908  64.472    13.563  63.633    14.224  62.799   
 14.889  61.970    15.560  61.147    16.236  60.329    16.918  59.517   
 17.605  58.711    18.298  57.911    18.997  57.116    19.703  56.328   
 20.414  55.546    21.132  54.771    21.857  54.003    22.589  53.242   
 23.327  52.488    24.073  51.741    24.826  51.002    25.587  50.270   
 26.356  49.547    27.133  48.832    27.918  48.126    28.713  47.428   
 29.516  46.740    30.328  46.061    31.150  45.392    31.982  44.734   
 32.825  44.086    33.678  43.449    34.543  42.823    35.419  42.209   
 36.307  41.608    37.208  41.020    38.122  40.445    39.050  39.884   
 39.992  39.338    40.949  38.808    41.922  38.293    42.912  37.796   
 43.919  37.317    44.945  36.857    45.991  36.417    47.057  35.998   
 48.145  35.602    49.257  35.230    50.393  34.884    51.556  34.565   
 52.748  34.276    53.971  34.019    55.227  33.796    56.520  33.611   
 57.852  33.466    59.227  33.367    60.650  33.317    62.127  33.323   
 63.593  33.386    65.004  33.500    66.368  33.662    67.688  33.865   
 68.968  34.107    70.212  34.383    71.423  34.692    72.602  35.031   
 73.754  35.397    74.879  35.789    75.979  36.205    77.056  36.644   
 78.111  37.103    79.146  37.583    80.161  38.081    81.158  38.597   
 82.138  39.130    83.102  39.679    84.049  40.243    84.982  40.822   
 85.901  41.415    86.806  42.020    87.698  42.639    88.577  43.270   
 89.445  43.912    90.301  44.566    91.147  45.230    91.981  45.905   
 92.806  46.589    93.620  47.284    94.425  47.987    95.000  48.500   
] scurve 
  9.399  65.369 moveto 
[   9.399  65.369     9.911  64.387    10.428  63.410    10.948  62.437   
 11.473  61.468    12.003  60.505    12.538  59.547    13.078  58.594   
 13.623  57.647    14.174  56.706    14.731  55.771    15.294  54.843   
 15.864  53.923    16.441  53.009    17.026  52.104    17.618  51.207   
 18.219  50.319    18.829  49.440    19.448  48.571    20.077  47.712   
 20.717  46.865    21.368  46.029    22.031  45.207    22.707  44.397   
 23.396  43.602    24.100  42.823    24.820  42.060    25.557  41.315   
 26.312  40.588    27.086  39.883    27.882  39.200    28.701  38.541   
 29.545  37.908    30.416  37.304    31.317  36.731    32.250  36.193   
 33.220  35.692    34.230  35.234    35.285  34.822    36.389  34.462   
 37.548  34.161    38.771  33.924    40.064  33.762    41.438  33.683   
 42.903  33.700    44.339  33.809    45.694  33.991    46.986  34.229   
 48.229  34.513    49.430  34.833    50.598  35.184    51.738  35.559   
 52.856  35.955    53.954  36.367    55.037  36.793    56.108  37.229   
 57.169  37.674    58.223  38.125    59.271  38.580    60.315  39.037   
 61.358  39.494    62.400  39.951    63.444  40.406    64.490  40.856   
 65.541  41.302    66.597  41.741    67.660  42.173    68.731  42.596   
 69.810  43.010    70.900  43.413    72.000  43.805    73.112  44.185   
 74.236  44.552    75.373  44.907    76.523  45.248    77.687  45.577   
 78.863  45.892    80.051  46.196    81.251  46.490    82.462  46.774   
 83.681  47.050    84.907  47.321    86.138  47.589    87.370  47.858   
 88.600  48.130    89.826  48.408    91.044  48.697    92.251  48.999   
 93.443  49.317    94.618  49.655    95.000  49.770   ] scurve 
  6.960  65.352 moveto 
[   6.960  65.352     7.318  64.214     7.679  63.079     8.042  61.946   
  8.406  60.815     8.773  59.687     9.142  58.561     9.514  57.438   
  9.889  56.318    10.266  55.201    10.646  54.087    11.029  52.977   
 11.416  51.870    11.806  50.767    12.200  49.669    12.598  48.574   
 13.000  47.484    13.406  46.399    13.817  45.318    14.233  44.244   
 14.654  43.174    15.080  42.111    15.513  41.055    15.952  40.005   
 16.397  38.962    16.850  37.927    17.311  36.901    17.779  35.883   
 18.257  34.875    18.744  33.878    19.241  32.891    19.749  31.917   
 20.270  30.955    20.803  30.008    21.351  29.077    21.914  28.162   
 22.495  27.267    23.094  26.392    23.715  25.541    24.360  24.716   
 25.032  23.921    25.735  23.160    26.474  22.438    27.254  21.763   
 28.083  21.142    28.972  20.589    29.934  20.118    30.991  19.755   
 32.174  19.535    33.533  19.520    34.920  19.759    36.109  20.160   
 37.171  20.665    38.146  21.245    39.055  21.882    39.912  22.563   
 40.728  23.281    41.509  24.028    42.262  24.801    42.990  25.595   
 43.698  26.407    44.387  27.235    45.061  28.077    45.721  28.930   
 46.370  29.794    47.008  30.667    47.637  31.548    48.258  32.435   
 48.873  33.328    49.481  34.226    50.085  35.127    50.684  36.032   
 51.280  36.940    51.874  37.849    52.465  38.759    53.055  39.670   
 53.645  40.580    54.235  41.490    54.825  42.399    55.416  43.305   
 56.009  44.209    56.605  45.109    57.204  46.005    57.807  46.897   
 58.414  47.784    59.027  48.664    59.646  49.537    60.271  50.402   
 60.905  51.259    61.547  52.105    62.199  52.941    62.862  53.764   
 63.537  54.574    64.227  55.368    64.931  56.146    65.653  56.904   
 66.395  57.641    67.158  58.355    67.946  59.040    68.763  59.695   
 69.612  60.314    70.499  60.892    71.430  61.422    72.413  61.895   
 73.457  62.299    74.579  62.619    75.795  62.835    77.133  62.915   
 78.631  62.813    79.955  62.562    81.159  62.208    82.277  61.777   
 83.328  61.289    84.326  60.754    85.283  60.182    86.205  59.580   
 87.099  58.953    87.969  58.306    88.820  57.642    89.655  56.966   
 90.478  56.278    91.290  55.583    92.094  54.882    92.893  54.178   
 93.689  53.472    94.484  52.767    95.000  52.310   ] scurve 
  0.113 setlinewidth
 61.562  31.394 moveto 
 85.562  39.273 lineto 
stroke 
 61.562  40.394 moveto 
 85.562  48.273 lineto 
stroke 
 61.562  58.394 moveto 
 85.562  66.273 lineto 
stroke
 30.219  45.298 moveto 
 53.219  31.172 lineto 
stroke 
 30.219  16.024 moveto 
 53.219  31.172 lineto 
stroke 
 30.219  35.540 moveto 
 53.219  31.172 lineto 
stroke 
 53.219  31.172 moveto 
  0.700 disk 
[   2.000   1.000 ] 0 setdash 
 74.000  26.800 moveto 
 74.000  62.200 lineto 
stroke 
 38.000  21.000 moveto 
 38.000  40.500 lineto 
stroke 
 19.000  57.000 moveto 
 19.000  33.500 lineto 
stroke 
[] 0 setdash 
 19.000  42.300 moveto 
  0.700   0.000 indent 
 19.000  41.800 moveto 
  0.700   0.000 indent 
 74.000  25.000 moveto 
  0.700  90.000 indent 
 38.000  28.500 moveto 
  0.700   0.000 indent 
 38.000  28.000 moveto 
  0.700   0.000 indent 
 74.000  53.300 moveto 
  0.700   0.000 indent 
 74.000  52.800 moveto 
  0.700   0.000 indent 
 74.000  39.800 moveto 
  1.400   0.000 cross 
 38.000  37.500 moveto 
  1.400   0.000 cross 
 19.000  53.500 moveto 
  1.400   0.000 cross}
\LabelTeX  103.000  21.200 $x$\ELTX
\LabelTeX   54.500  68.000 $y$\ELTX
\LabelTeX   73.500  21.000 $x_0$\ELTX
\EndFig
\vskip-6mm

\subsection 1.5. \'Equations se ramenant \`a des \'equations lin\'eaires|

\noindent {\tenpointbf a) \'Equations de Bernoulli}
\medskip

Ce sont les \'equations de la forme
$$
\frac{dy}{dx}=p(x)y+q(x)y^{\alpha},\quad \alpha\in\bR\setminus\{1\},\leqno {\rm (E)}
$$
avec $p,q:I\rightarrow \bR$ continues (pour $\alpha=1$, (E) est lin\'eaire).

On se place dans le demi-plan sup\'erieur $U=\bR\;\times\ ]0,+\infty[
=\{(x,y)\,;\;y>0\}$. En multipliant par $y^{-\alpha}$, on obtient
$$
{\rm (E)}\quad\Leftrightarrow\quad y^{-\alpha}\frac{dy}{dx}
=p(x)y^{1-\alpha}+q(x)
$$
Posons $z=y^{1-\alpha}$~; alors $\dfrac{dz}{dx}=
(1-\alpha)y^{-\alpha}\dfrac{dy}{dx}$, d'o\`u
$$
{\rm (E)}\quad\Leftrightarrow\quad\frac{1}{1-\alpha}\ \frac{dz}{dx}=p(x)z+q(x)
$$
On est donc ramen\'e \`a une \'equation lin\'eaire en $z$.

\bigskip
\noindent {\tenpointbf b) \'Equations de Riccati}
\medskip

Ce sont les \'equations de la forme
$$
y'=a(x)y^2+b(x)y+c(x)\leqno{\rm (E)}
$$
avec $a,b,c:I\rightarrow\bR$ continues, c'est-\`a-dire que $f(x,y)$ est un
polyn\^ome de degr\'e $\leq 2$ en $y$. Montrons que l'on sait r\'esoudre (E)
d\`es que l'on conna\^{\i}t une solution particuli\`ere $y_{(1)}$. 
Posons $y=y_{(1)}+z$. Il vient
$$
\eqalign{
y'_{(1)}+z'&=a(x)(y_{(1)}^2+2y_{(1)}z+z^2)+b(x)(y_{(1)}+z)+c(x)\cr
&=a(x)y^2_{(1)}+b(x)y_{(1)}+c(x)+(2a(x)y_{(1)}+b(x))z+a(x)z^2.\cr
}
$$
Comme $y'_{(1)}$ se simplifie, on en d\'eduit
$$
z'=(2a(x)y_{(1)}(x)+b(x))+a(x)z^2.
$$
C'est une \'equation lin\'eaire de Bernoulli avec $\alpha=2$. On la ram\`ene \`a une
\'equation lin\'eaire en posant $w=z^{1-\alpha}=\dfrac{1}{z}$.

\bigskip
\state Exemple -- |  Soit l'\'equation $(1-x^3)y'+x^2y+y^2-2x=0$.
\medskip

On remarque que $y_{(1)}(x)=x^2$ est solution particuli\`ere. En posant $y=x^2+z$ on se
ram\`ene \`a 
$$
(1-x^3)z'+3x^2z+z^2=0
$$
puis, apr\`es division par $z^2$, \`a
$$
-(1-x^3)w'+3x^2w+1=0\quad{\rm avec}\quad w=\frac{1}{z},
$$
soit
$$
w'=\frac{3x^2}{1-x^3}\ w+\frac{1}{1-x^3},\quad{\rm si}\quad x\not=1.
$$
L'\'equation lin\'eaire sans second membre $\dfrac{w'}{w}=\dfrac{3x^2}{1-x^3}$ donne
$$
\ln\,|w|=-\ln\,|1-x^3|+C,\quad\hbox{d'o\`u}\quad w=\frac{\lambda}{1-x^3}.
$$
La m\'ethode de variation des constantes conduit \`a 
$$
\frac{\lambda'}{1-x^3}=\frac{1}{1-x^3}\quad{\rm soit}\quad \lambda'=1,\quad \lambda(x)=x.
$$
La solution g\'en\'erale de l'\'equation lin\'eaire compl\`ete est donc
$$
w(x)=\frac{x+\lambda}{1-x^3},
$$
d'o\`u\quad $y=x^2+z=x^2+\dfrac{1}{w}=x^2+\dfrac{1-x^3}{x+\lambda}$, soit encore
$$
y(x)=\dfrac{\lambda x^2+1}{x+\lambda}=\lambda x-\lambda^2+\dfrac{1+\lambda^3}{x+\lambda}.
$$
Pour $\lambda=-1$, on obtient la droite $y=-x-1$. Pour $\lambda\neq-1$, il s'agit d'une
hyperbole $(y-\lambda x+\lambda^2)(x+\lambda)=1+\lambda^3$, admettant pour asymptotes les
droites $x=-\lambda$ et $y=\lambda x-\lambda^2$. La solution singuli\`ere
$y_{(1)}(x)=x^2$ est la solution limite obtenue quand $\vert\lambda\vert$ 
tend vers $+\infty$.

\InsertFig 16.000 81.000 {
1 mm unit
  0.000  42.000 moveto 
100.000   0.000 vector 
 50.000   0.000 moveto 
 70.000  90.000 vector 
 50.000  52.000 moveto 
  0.700   0.000 indent 
 60.000  42.000 moveto 
  0.700  90.000 indent 
gsave 
  5.000   5.000 moveto 
 90.000  60.000 rect  clip  newpath 
 10.000  72.000 moveto 
 80.000   2.000 lineto 
stroke 
[   2.000   1.000 ] 0 setdash 
 60.000   2.000 moveto 
 60.000  67.000 lineto 
stroke 
[ ] 0 setdash 
 70.000  62.000 moveto 
[  70.000  62.000    68.838  60.686    67.884  59.629    67.084  58.762   
 66.348  57.977    65.608  57.205    64.848  56.429    64.073  55.657   
 63.294  54.904    62.635  54.284    62.132  53.823    61.739  53.471   
 61.429  53.197    61.180  52.981    60.978  52.809    60.813  52.669   
 60.678  52.556    60.567  52.463    60.475  52.386    60.398  52.323   
 60.334  52.271    60.281  52.227    60.236  52.191    60.199  52.160   
 60.167  52.135    60.141  52.114    60.119  52.096    60.100  52.081   
 60.085  52.068    60.071  52.057    60.060  52.048    60.051  52.041   
 60.043  52.034    60.036  52.029    60.031  52.025    60.026  52.021   
 60.022  52.017    60.018  52.015    60.016  52.012    60.013  52.011   
 60.011  52.009    60.009  52.008    60.008  52.006    60.007  52.005   
 60.006  52.005    60.005  52.004    60.004  52.003    60.003  52.003   
 60.003  52.002    60.002  52.002    60.002  52.002    60.002  52.001   
 60.001  52.001    60.001  52.001    60.001  52.001   ] scurve 
 70.000  62.000 moveto 
[  70.000  62.000    71.452  63.679    73.327  65.904   ] scurve 
 90.000  62.000 moveto 
[  90.000  62.000    88.216  61.212    86.654  60.529    85.273  59.932   
 84.039  59.404    82.930  58.934    81.925  58.512    81.010  58.132   
 80.171  57.787    79.399  57.473    78.685  57.186    78.024  56.922   
 77.407  56.679    76.832  56.454    76.293  56.245    75.787  56.052   
 75.311  55.871    74.862  55.702    74.438  55.544    74.036  55.396   
 73.654  55.257    73.292  55.126    72.947  55.003    72.618  54.886   
 72.304  54.776    72.005  54.672    71.718  54.573    71.412  54.469   
 71.086  54.359    70.776  54.256    70.481  54.159    70.199  54.068   
 69.930  53.981    69.674  53.900    69.428  53.823    69.192  53.750   
 68.967  53.681    68.750  53.615    68.542  53.553    68.342  53.494   
 68.149  53.438    67.964  53.384    67.785  53.333    67.612  53.285   
 67.446  53.238    67.285  53.194    67.129  53.152    66.979  53.111   
 66.834  53.073    66.693  53.036    66.556  53.000    66.424  52.966   
 66.296  52.934    66.171  52.902    66.050  52.873    65.933  52.844   
 65.814  52.815    65.676  52.782    65.543  52.751    65.415  52.721   
 65.290  52.693    65.170  52.665    65.054  52.640    64.941  52.615   
 64.832  52.591    64.726  52.569    64.624  52.547    64.524  52.527   
 64.428  52.507    64.334  52.488    64.243  52.470    64.155  52.453   
 64.069  52.437    63.985  52.421    63.904  52.406    63.825  52.391   
 63.748  52.377    63.673  52.364    63.600  52.351    63.529  52.338   
 63.460  52.327    63.393  52.315    63.327  52.304    63.263  52.294   
 63.201  52.284    63.140  52.274    63.080  52.264    63.022  52.255   
 62.965  52.247    62.910  52.238    62.856  52.230    62.803  52.223   
 62.752  52.215    62.701  52.208    62.652  52.201    62.604  52.195   
 62.557  52.188    62.511  52.182    62.466  52.176    62.422  52.170   
 62.379  52.165    62.337  52.159    62.296  52.154    62.255  52.149   
 62.216  52.144    62.177  52.140    62.139  52.135    62.102  52.131   
 62.066  52.127    62.031  52.123    61.996  52.119    61.962  52.115   
 61.928  52.111    61.895  52.108    61.863  52.104    61.832  52.101   
 61.801  52.098    61.771  52.095    61.741  52.092    61.712  52.089   
 61.684  52.086    61.656  52.084    61.628  52.081    61.601  52.079   
 61.575  52.076    61.549  52.074    61.524  52.072    61.499  52.069   
 61.474  52.067    61.450  52.065    61.427  52.063    61.403  52.061   
 61.381  52.059    61.358  52.058    61.337  52.056    61.315  52.054   
 61.294  52.053    61.273  52.051    61.253  52.049    61.233  52.048   
 61.213  52.047    61.194  52.045    61.175  52.044    61.156  52.042   
 61.138  52.041    61.120  52.040    61.103  52.039    61.085  52.038   
 61.068  52.037    61.051  52.035    61.035  52.034    61.019  52.033   
 61.003  52.032    60.987  52.031    60.972  52.031    60.957  52.030   
 60.942  52.029    60.927  52.028    60.913  52.027    60.899  52.026   
 60.885  52.026    60.871  52.025    60.858  52.024    60.845  52.023   
 60.832  52.023    60.819  52.022    60.807  52.021    60.794  52.021   
 60.782  52.020    60.770  52.020    60.758  52.019    60.747  52.018   
 60.736  52.018    60.724  52.017    60.713  52.017    60.703  52.016   
 60.692  52.016    60.682  52.016    60.671  52.015    60.661  52.015   
 60.651  52.014    60.641  52.014    60.632  52.013    60.622  52.013   
 60.613  52.013    60.604  52.012    60.595  52.012    60.586  52.012   
 60.577  52.011    60.568  52.011   ] scurve 
 90.000  62.000 moveto 
[  90.000  62.000    92.065  62.924    94.493  64.023    97.404  65.354   
] scurve 
 84.281  49.144 moveto 
[  84.281  49.144    82.822  49.074    81.707  49.027    80.702  48.991   
 79.789  48.963    78.956  48.943    78.192  48.929    77.488  48.920   
 76.836  48.915    76.231  48.914    75.667  48.916    75.140  48.921   
 74.647  48.928    74.183  48.938    73.746  48.949    73.334  48.962   
 72.944  48.976    72.574  48.991    72.224  49.008    71.891  49.025   
 71.574  49.043    71.271  49.062    70.983  49.082    70.707  49.102   
 70.443  49.123    70.190  49.144    69.948  49.165    69.715  49.187   
 69.491  49.208    69.276  49.230    69.069  49.253    68.870  49.275   
 68.678  49.298    68.492  49.320    68.313  49.343    68.140  49.366   
 67.973  49.388    67.811  49.411    67.654  49.434    67.502  49.456   
 67.355  49.479    67.213  49.502    67.075  49.524    66.940  49.546   
 66.810  49.569    66.684  49.591    66.561  49.613    66.441  49.635   
 66.325  49.657    66.212  49.679    66.101  49.701    65.994  49.722   
 65.890  49.744    65.788  49.765    65.689  49.786    65.592  49.807   
 65.498  49.828    65.406  49.849    65.316  49.869    65.228  49.890   
 65.142  49.910    65.059  49.930    64.977  49.950    64.897  49.970   
 64.819  49.989    64.743  50.009    64.668  50.028    64.595  50.047   
 64.524  50.066    64.454  50.085    64.385  50.104    64.318  50.123   
 64.253  50.141    64.189  50.159    64.126  50.177    64.064  50.195   
 64.004  50.213    63.945  50.230    63.886  50.248    63.830  50.265   
 63.774  50.282    63.719  50.299    63.666  50.316    63.613  50.333   
 63.561  50.349    63.511  50.366    63.461  50.382    63.412  50.398   
 63.364  50.414    63.317  50.430    63.271  50.445    63.225  50.461   
 63.181  50.476    63.137  50.491    63.094  50.506    63.052  50.521   
 63.010  50.536    62.970  50.550    62.929  50.565    62.890  50.579   
 62.851  50.593    62.813  50.608    62.776  50.621    62.739  50.635   
 62.703  50.649    62.667  50.662    62.632  50.676    62.597  50.689   
 62.564  50.702    62.530  50.715    62.497  50.728    62.465  50.741   
 62.433  50.754    62.402  50.766    62.371  50.778    62.341  50.791   
 62.311  50.803    62.281  50.815    62.252  50.827    62.224  50.839   
 62.196  50.850    62.168  50.862    62.141  50.873    62.114  50.885   
 62.088  50.896    62.062  50.907    62.036  50.918    62.011  50.929   
 61.986  50.940    61.961  50.951    61.937  50.961    61.913  50.972   
 61.890  50.982    61.867  50.992    61.844  51.002    61.818  51.014   
 61.790  51.027    61.763  51.039    61.737  51.051    61.710  51.063   
 61.685  51.075    61.659  51.086    61.635  51.098    61.610  51.109   
 61.586  51.121    61.563  51.132    61.539  51.143    61.517  51.153   
 61.494  51.164    61.472  51.175    61.451  51.185    61.429  51.195   
 61.408  51.206    61.388  51.216    61.368  51.226    61.348  51.235   
 61.328  51.245    61.309  51.255    61.290  51.264    61.271  51.273   
 61.253  51.283    61.235  51.292    61.217  51.301    61.200  51.310   
 61.183  51.318    61.166  51.327    61.149  51.335    61.133  51.344   
 61.117  51.352    61.101  51.360    61.085  51.368    61.070  51.376   
 61.055  51.384    61.040  51.392    61.025  51.400    61.011  51.408   
 60.996  51.415    60.982  51.422    60.969  51.430    60.955  51.437   
 60.942  51.444    60.928  51.451    60.915  51.458    60.903  51.465   
 60.890  51.472    60.878  51.479    60.866  51.485    60.854  51.492   
 60.842  51.498    60.830  51.505   ] scurve 
[  60.830  51.505    60.819  51.511    60.807  51.517    60.796  51.523   
 60.785  51.529    60.774  51.535    60.764  51.541    60.753  51.547   
 60.743  51.553    60.733  51.558    60.723  51.564    60.713  51.570   
 60.703  51.575    60.693  51.580    60.684  51.586    60.675  51.591   
 60.665  51.596    60.656  51.601    60.647  51.606    60.639  51.611   
 60.630  51.616    60.621  51.621    60.613  51.626    60.605  51.631   
 60.597  51.635    60.588  51.640    60.580  51.645   ] scurve 
 84.281  49.144 moveto 
[  84.281  49.144    85.764  49.227    87.058  49.307    88.507  49.403   
 90.112  49.517    91.582  49.628    93.230  49.759    95.094  49.915   
 96.895  50.072   ] scurve 
 77.135  42.000 moveto 
[  77.135  42.000    75.945  42.350    75.043  42.628    74.234  42.887   
 73.504  43.130    72.841  43.360    72.235  43.577    71.680  43.783   
 71.168  43.980    70.694  44.168    70.255  44.348    69.845  44.520   
 69.463  44.686    69.105  44.846    68.770  45.000    68.454  45.149   
 68.156  45.292    67.875  45.432    67.609  45.567    67.358  45.697   
 67.119  45.824    66.891  45.948    66.675  46.068    66.469  46.185   
 66.273  46.298    66.085  46.409    65.906  46.517    65.735  46.622   
 65.570  46.724    65.413  46.824    65.261  46.922    65.116  47.017   
 64.977  47.111    64.842  47.202    64.713  47.291    64.589  47.378   
 64.469  47.463    64.354  47.546    64.242  47.628    64.134  47.708   
 64.031  47.786    63.930  47.862    63.833  47.937    63.739  48.011   
 63.648  48.083    63.560  48.153    63.475  48.222    63.392  48.290   
 63.312  48.356    63.235  48.421    63.159  48.485    63.086  48.547   
 63.015  48.609    62.947  48.669    62.880  48.728    62.815  48.786   
 62.752  48.843    62.690  48.898    62.631  48.953    62.572  49.007   
 62.516  49.059    62.461  49.111    62.408  49.162    62.355  49.211   
 62.305  49.260    62.255  49.308    62.207  49.355    62.160  49.402   
 62.115  49.447    62.070  49.492    62.027  49.535    61.984  49.578   
 61.943  49.621    61.903  49.662    61.863  49.703    61.825  49.743   
 61.788  49.782    61.751  49.821    61.715  49.858    61.681  49.896   
 61.647  49.932    61.613  49.968    61.581  50.003    61.549  50.038   
 61.518  50.072    61.488  50.105    61.458  50.138    61.429  50.170   
 61.401  50.202    61.374  50.233    61.346  50.264    61.320  50.294   
 61.294  50.323    61.269  50.352    61.244  50.381    61.220  50.409   
 61.196  50.436    61.173  50.463    61.151  50.490    61.128  50.516   
 61.107  50.542    61.086  50.567    61.065  50.592    61.044  50.616   
 61.024  50.640    61.005  50.663    60.986  50.686    60.967  50.709   
 60.949  50.731    60.931  50.753    60.913  50.775    60.896  50.796   
 60.879  50.817    60.863  50.837    60.847  50.857    60.831  50.877   
 60.815  50.896    60.800  50.915    60.785  50.934    60.770  50.952   
 60.756  50.970    60.742  50.988    60.728  51.006    60.715  51.023   
 60.702  51.040    60.689  51.056    60.676  51.072    60.664  51.088   
 60.651  51.104    60.639  51.120    60.628  51.135    60.616  51.150   
 60.605  51.164    60.594  51.179    60.583  51.193    60.572  51.207   
 60.562  51.221    60.552  51.234    60.542  51.247    60.532  51.260   
 60.522  51.273    60.513  51.285    60.503  51.298    60.494  51.310   
 60.485  51.322    60.477  51.333    60.468  51.345    60.459  51.356   
 60.451  51.367    60.443  51.378    60.435  51.389    60.427  51.399   
 60.420  51.410    60.412  51.420    60.405  51.430    60.397  51.440   
 60.390  51.449    60.383  51.459    60.376  51.468    60.370  51.477   
 60.363  51.486    60.357  51.495    60.350  51.504    60.344  51.513   
 60.338  51.521    60.332  51.529    60.326  51.537    60.320  51.545   
 60.314  51.553    60.309  51.561   ] scurve 
 77.135  42.000 moveto 
[  77.135  42.000    78.353  41.661    79.424  41.376    80.609  41.072   
 81.693  40.803    82.906  40.513    84.274  40.197    85.505  39.922   
 86.881  39.623    88.436  39.296    89.835  39.009    91.402  38.697   
 93.138  38.359    94.729  38.057    96.513  37.726   ] scurve 
 64.281  42.000 moveto 
[  64.281  42.000    63.787  42.767    63.495  43.248    63.232  43.702   
 62.994  44.131    62.778  44.536    62.581  44.921    62.400  45.285   
 62.235  45.631    62.084  45.959    61.945  46.271    61.830  46.535   
 61.733  46.762    61.643  46.980    61.557  47.190    61.477  47.391   
 61.402  47.584    61.331  47.769    61.263  47.946    61.200  48.117   
 61.140  48.281    61.084  48.438    61.030  48.588    60.980  48.733   
 60.932  48.872    60.887  49.005    60.844  49.132    60.803  49.254   
 60.764  49.372    60.728  49.484    60.693  49.592    60.660  49.695   
 60.629  49.794    60.599  49.889    60.571  49.980    60.544  50.067   
 60.518  50.151    60.494  50.231    60.471  50.308    60.449  50.381   
 60.428  50.451    60.408  50.519    60.390  50.583    60.372  50.645   
 60.354  50.704    60.338  50.760    60.322  50.815    60.308  50.866   
 60.294  50.916    60.280  50.963    60.267  51.009    60.255  51.052   
 60.243  51.094    60.232  51.134    60.222  51.172    60.214  51.201   
 60.206  51.229    60.198  51.257    60.191  51.283    60.184  51.308   
 60.177  51.333    60.171  51.356    60.165  51.379    60.159  51.401   
 60.153  51.422    60.147  51.443    60.142  51.463    60.137  51.482   
 60.132  51.500    60.127  51.518    60.123  51.535    60.118  51.552   
 60.114  51.567    60.110  51.583    60.106  51.598    60.102  51.612   
 60.098  51.626    60.095  51.639    60.091  51.652    60.088  51.664   
 60.085  51.676    60.082  51.688    60.079  51.699    60.076  51.710   
 60.073  51.720    60.071  51.730    60.068  51.740    60.066  51.749   
 60.063  51.758    60.061  51.767   ] scurve 
 64.281  42.000 moveto 
[  64.281  42.000    64.869  41.159    65.332  40.540    65.862  39.871   
 66.474  39.147    67.161  38.384    67.821  37.694    68.584  36.939   
 69.478  36.107    70.416  35.283    71.404  34.459    72.549  33.552   
 73.649  32.721    74.944  31.784    76.193  30.916    77.628  29.952   
 79.008  29.055    80.590  28.057    82.070  27.146    83.796  26.109   
 85.404  25.162    87.273  24.082    89.049  23.072    91.062  21.944   
 92.968  20.891    95.117  19.717    97.190  18.597   ] scurve 
 67.138  24.862 moveto 
[  67.138  24.862    66.103  26.352    65.277  27.675    64.604  28.878   
 64.044  29.995    63.573  31.044    63.146  32.108    62.726  33.292   
 62.373  34.422    62.044  35.627    61.737  36.919    61.482  38.150   
 61.270  39.320    61.090  40.430    60.957  41.348    60.850  42.147   
 60.772  42.771    60.703  43.353    60.641  43.909    60.594  44.345   
 60.552  44.753    60.513  45.145    60.477  45.520    60.444  45.879   
 60.416  46.185    60.393  46.449    60.371  46.704    60.350  46.948   
 60.330  47.184    60.312  47.409    60.295  47.626    60.278  47.834   
 60.263  48.033    60.248  48.224    60.235  48.407    60.222  48.582   
 60.211  48.722    60.202  48.852    60.193  48.977    60.184  49.097   
 60.176  49.213    60.169  49.324    60.161  49.432    60.154  49.536   
 60.147  49.635    60.141  49.731    60.134  49.824    60.129  49.913   
 60.123  49.998    60.118  50.080    60.112  50.159    60.107  50.235   
 60.103  50.308    60.098  50.378    60.094  50.445    60.090  50.510   
 60.086  50.572    60.082  50.632    60.078  50.689    60.075  50.744   
 60.072  50.796    60.069  50.847    60.066  50.895    60.063  50.942   
 60.060  50.987    60.057  51.029    60.055  51.070    60.052  51.110   
 60.050  51.147    60.048  51.184    60.046  51.218    60.044  51.252   
 60.042  51.283    60.040  51.314    60.038  51.343    60.037  51.371   
 60.035  51.398    60.034  51.424    60.032  51.448    60.031  51.472   
 60.029  51.495    60.028  51.516    60.027  51.537    60.026  51.557   
 60.025  51.576    60.024  51.594    60.022  51.612    60.022  51.628   
] scurve 
 67.138  24.862 moveto 
[  67.138  24.862    68.475  23.123    70.279  20.996    72.870  18.205   
 76.032  15.039    78.909  12.291    81.665   9.732    84.437   7.207   
 87.149   4.771    89.864   2.358   ] scurve 
 69.994   7.724 moveto 
[  69.994   7.724    69.125   8.445    68.506   8.932    67.954   9.343   
 67.456   9.689    67.007   9.978    66.598  10.216    66.224  10.410   
 65.881  10.563    65.566  10.678    65.274  10.758    65.052  10.799   
 64.847  10.819    64.654  10.820    64.472  10.803    64.299  10.767   
 64.135  10.713    63.979  10.643    63.794  10.528    63.322  10.048   
 62.929   9.349    62.722   8.818    62.573   8.337    62.442   7.833   
 62.343   7.389    62.249   6.906    62.162   6.392    62.093   5.943   
 62.027   5.464    61.964   4.953    61.902   4.409   ] scurve 
 69.994   7.724 moveto 
[  69.994   7.724    71.120   6.740    72.387   5.585    73.945   4.116   
] scurve 
 50.000   9.153 moveto 
[  50.000   9.153    50.107   7.964    50.168   7.239    50.229   6.484   
 50.286   5.753    50.336   5.099    50.385   4.422   ] scurve 
 50.000   9.153 moveto 
[  50.000   9.153    49.893  10.266    49.832  10.875    49.771  11.462   
 49.709  12.028    49.648  12.574    49.586  13.102    49.525  13.611   
 49.463  14.104    49.402  14.581    49.340  15.043    49.279  15.490   
 49.217  15.923    49.156  16.344    49.094  16.752    49.033  17.148   
 48.971  17.532    48.910  17.906    48.848  18.270    48.787  18.623   
 48.725  18.968    48.663  19.303    48.602  19.629    48.540  19.947   
 48.478  20.257    48.417  20.560    48.355  20.855    48.293  21.143   
 48.232  21.425    48.170  21.700    48.108  21.968    48.046  22.231   
 47.984  22.488    47.922  22.739    47.860  22.985    47.798  23.226   
 47.736  23.461    47.674  23.692    47.611  23.919    47.549  24.141   
 47.487  24.358    47.424  24.571    47.362  24.781    47.299  24.986   
 47.236  25.188    47.173  25.386    47.110  25.581    47.047  25.772   
 46.984  25.960    46.921  26.144    46.858  26.326    46.794  26.505   
 46.731  26.681    46.667  26.854    46.603  27.024    46.540  27.192   
 46.475  27.357    46.411  27.520    46.347  27.680    46.282  27.838   
 46.218  27.994    46.153  28.148    46.088  28.300    46.022  28.449   
 45.957  28.597    45.891  28.743    45.826  28.887    45.760  29.029   
 45.693  29.169    45.627  29.308    45.560  29.445    45.493  29.580   
 45.426  29.714    45.358  29.847    45.291  29.978    45.223  30.108   
 45.154  30.236    45.086  30.363    45.017  30.489    44.948  30.613   
 44.878  30.737    44.808  30.859    44.738  30.980    44.667  31.100   
 44.596  31.219    44.525  31.337    44.453  31.454    44.373  31.584   
 44.282  31.728    44.190  31.870    44.098  32.011    44.005  32.151   
 43.911  32.290    43.817  32.428    43.721  32.565    43.625  32.701   
 43.521  32.845    43.398  33.012    43.274  33.179    43.148  33.344   
 43.020  33.508    42.890  33.671    42.741  33.856    42.573  34.057   
 42.402  34.259    42.228  34.459    42.004  34.709    41.774  34.959   
 41.487  35.260    41.142  35.610    40.675  36.060    39.859  36.795   
 38.850  37.626    37.651  38.523    36.529  39.293    35.581  39.900   
 34.706  40.431    33.857  40.923    33.027  41.385    32.226  41.815   
 31.384  42.252    30.612  42.641    29.770  43.053    28.950  43.444   
 28.114  43.833    27.197  44.249    26.367  44.618    25.447  45.018   
 24.497  45.423    23.588  45.804    22.581  46.218    21.563  46.630   
 20.572  47.025    19.473  47.456    18.341  47.893    17.260  48.306   
 16.063  48.756    14.805  49.223    13.627  49.656    12.325  50.129   
 10.876  50.650     9.590  51.108     8.170  51.609     6.592  52.160   
  5.094  52.679     3.542  53.212   ] scurve 
 32.862  29.147 moveto 
[  32.862  29.147    34.820  28.568    36.324  27.845    37.356  27.144   
 38.060  26.536    38.626  25.951    39.086  25.399    39.492  24.842   
 39.817  24.344    40.127  23.817    40.423  23.261    40.662  22.772   
 40.882  22.281    41.096  21.767    41.303  21.227    41.505  20.660   
 41.673  20.152    41.828  19.656    41.979  19.138    42.127  18.598   
 42.273  18.033    42.416  17.441    42.556  16.821    42.667  16.303   
 42.777  15.765    42.885  15.204    42.992  14.620    43.097  14.011   
 43.202  13.375    43.305  12.711    43.407  12.016    43.493  11.396   
 43.574  10.793    43.653  10.166    43.732   9.514    43.811   8.836   
 43.889   8.129    43.966   7.391    44.043   6.622    44.119   5.818   
 44.190   5.039    44.250   4.341   ] scurve 
 32.862  29.147 moveto 
[  32.862  29.147    30.712  29.470    29.004  29.568    27.416  29.569   
 25.775  29.498    24.128  29.370    22.358  29.182    20.434  28.930   
 18.536  28.642    16.381  28.277    14.222  27.878    11.860  27.411   
  9.352  26.886     6.713  26.309     3.787  25.643   ] scurve 
 32.862  19.150 moveto 
[  32.862  19.150    34.004  18.674    34.807  18.180    35.418  17.693   
 35.975  17.143    36.416  16.624    36.802  16.096    37.166  15.525   
 37.511  14.911    37.785  14.362    38.036  13.805    38.278  13.215   
 38.511  12.592    38.736  11.931    38.937  11.284    39.106  10.698   
 39.271  10.083    39.432   9.439    39.588   8.763    39.741   8.054   
 39.891   7.308    40.034   6.540    40.149   5.884    40.262   5.201   
 40.373   4.488   ] scurve 
 32.862  19.150 moveto 
[  32.862  19.150    31.251  19.495    29.338  19.558    26.787  19.260   
 23.809  18.559    20.844  17.619    17.624  16.418    14.142  14.976   
 10.315  13.271     6.110  11.299     1.498   9.049   ] scurve 
  7.156  39.144 moveto 
[   7.156  39.144     8.870  39.065    10.297  38.994    11.587  38.924   
 12.760  38.855    13.834  38.787    14.822  38.720    15.735  38.655   
 16.581  38.590    17.369  38.527    18.106  38.464    18.796  38.402   
 19.444  38.342    20.055  38.281    20.632  38.222    21.178  38.164   
 21.695  38.106    22.187  38.049    22.655  37.992    23.101  37.936   
 23.527  37.881    23.935  37.826    24.325  37.772    24.699  37.718   
 25.058  37.665    25.403  37.613    25.749  37.558    26.146  37.494   
 26.526  37.430    26.889  37.366    27.237  37.304    27.570  37.242   
 27.941  37.170    28.322  37.094    28.686  37.019    29.034  36.945   
 29.366  36.871    29.749  36.783    30.126  36.692    30.486  36.603   
 30.828  36.514    31.221  36.408    31.607  36.300    31.973  36.192   
 32.374  36.069    32.783  35.937    33.177  35.804    33.630  35.643   
 34.072  35.476    34.565  35.278    35.114  35.042    35.728  34.755   
 36.557  34.325    37.539  33.740    38.389  33.151    39.138  32.553   
 39.810  31.941    40.419  31.309    40.968  30.664    41.387  30.117   
 41.768  29.568    42.071  29.094    42.363  28.600    42.640  28.095   
 42.859  27.667    43.073  27.222    43.282  26.760    43.487  26.277   
 43.669  25.820    43.827  25.402    43.983  24.968    44.137  24.517   
 44.288  24.047    44.438  23.557    44.583  23.056    44.700  22.630   
 44.817  22.189    44.932  21.731    45.046  21.255    45.160  20.759   
 45.272  20.243    45.384  19.704    45.493  19.153    45.582  18.685   
 45.670  18.200    45.758  17.697    45.845  17.174    45.932  16.630   
 46.018  16.064    46.105  15.475    46.190  14.860    46.276  14.218   
 46.347  13.659    46.416  13.104    46.483  12.529    46.551  11.932   
 46.619  11.311    46.686  10.665    46.754   9.993    46.821   9.293   
 46.888   8.563    46.955   7.801    47.013   7.108    47.066   6.449   
 47.120   5.765    47.173   5.055    47.226   4.318   ] scurve 
  7.156  39.144 moveto 
[   7.156  39.144     5.210  39.221     3.539  39.280   ] scurve 
 28.578   9.153 moveto 
[  28.578   9.153    30.067   9.015    31.149   8.701    31.912   8.337   
 32.600   7.881    33.226   7.338    33.698   6.830    34.130   6.277   
 34.535   5.670    34.917   5.007    35.251   4.340   ] scurve 
 28.578   9.153 moveto 
[  28.578   9.153    26.411   8.927    23.731   8.185    20.034   6.644   
 15.879   4.478   ] scurve 
 50.000  42.000 moveto 
 28.578  60.566 moveto 
[  28.578  60.566    29.208  59.622    29.713  58.871    30.187  58.171   
 30.632  57.517    31.051  56.904    31.448  56.327    31.824  55.784   
 32.181  55.271    32.520  54.785    32.844  54.325    33.153  53.888   
 33.449  53.471    33.732  53.074    34.003  52.696    34.264  52.334   
 34.515  51.987    34.757  51.655    34.990  51.336    35.215  51.029   
 35.433  50.735    35.643  50.451    35.847  50.177    36.044  49.914   
 36.235  49.659    36.421  49.412    36.602  49.174    36.777  48.944   
 36.948  48.720    37.114  48.504    37.276  48.294    37.434  48.090   
 37.588  47.891    37.738  47.699    37.885  47.511    38.029  47.329   
 38.169  47.151    38.306  46.978    38.441  46.810    38.572  46.645   
 38.701  46.485    38.828  46.328    38.965  46.158    39.117  45.972   
 39.265  45.791    39.409  45.615    39.550  45.444    39.689  45.278   
 39.827  45.111    39.993  44.914    40.154  44.723    40.312  44.537   
 40.466  44.357    40.624  44.174    40.808  43.962    40.987  43.756   
 41.161  43.558    41.351  43.344    41.558  43.113    41.760  42.889   
 41.991  42.636    42.229  42.379    42.507  42.083    42.816  41.758   
 43.196  41.367    43.736  40.829    44.485  40.114    45.346  39.347   
 46.165  38.675    46.871  38.145    47.426  37.766    47.919  37.458   
 48.364  37.204    48.755  37.003    49.145  36.823    49.534  36.665   
 49.923  36.531    50.313  36.421    50.631  36.352    50.943  36.302   
 51.254  36.271    51.564  36.260    51.874  36.270    52.183  36.302   
 52.490  36.357    52.796  36.436    53.100  36.540    53.400  36.669   
 53.805  36.888    54.513  37.407    55.348  38.275    56.118  39.368   
 56.811  40.638    57.417  42.018    57.936  43.431    58.266  44.457   
 58.497  45.244    58.676  45.890    58.808  46.392    58.926  46.858   
 59.025  47.266    59.105  47.600    59.178  47.917    59.246  48.216   
 59.309  48.497    59.360  48.729    59.403  48.931    59.444  49.122   
 59.482  49.303    59.517  49.475    59.551  49.636    59.582  49.789   
 59.611  49.933    59.638  50.069    59.663  50.196    59.686  50.316   
 59.704  50.408    59.721  50.494    59.737  50.576    59.752  50.653   
 59.766  50.727    59.779  50.797    59.792  50.863    59.804  50.926   
 59.815  50.985    59.825  51.042    59.835  51.095    59.845  51.145   
 59.854  51.193    59.862  51.238    59.870  51.281    59.877  51.322   
 59.884  51.360    59.891  51.396    59.897  51.430    59.903  51.462   
 59.909  51.493    59.914  51.522    59.919  51.549    59.924  51.575   
 59.928  51.599    59.932  51.622    59.936  51.643    59.940  51.664   
 59.943  51.683    59.947  51.701    59.950  51.718    59.953  51.734   
 59.955  51.750    59.958  51.764   ] scurve 
 28.578  60.566 moveto 
[  28.578  60.566    27.970  61.479    27.466  62.242    26.929  63.058   
 26.354  63.936    25.738  64.882    25.199  65.715   ] scurve 
 50.000  42.000 moveto 
[  50.000  42.000    52.750  42.755    55.310  44.819    57.027  46.939   
 57.723  47.967    58.158  48.657    58.466  49.169    58.702  49.575   
 58.887  49.901    59.049  50.190    59.177  50.423    59.276  50.607   
 59.364  50.771    59.442  50.917    59.511  51.048    59.572  51.164   
 59.626  51.267    59.673  51.357    59.714  51.437    59.745  51.498   
 59.771  51.549    59.795  51.595    59.816  51.637    59.835  51.674   
 59.852  51.708    59.868  51.738    59.881  51.765    59.894  51.789   
 59.905  51.811    59.915  51.831    59.924  51.849    59.932  51.864   
 59.939  51.879    59.945  51.891    59.951  51.903    59.956  51.913   
 59.961  51.922    59.965  51.930    59.969  51.937    59.972  51.944   
] scurve 
 50.000  42.000 moveto 
[  50.000  42.000    47.272  42.743    46.008  43.591    45.247  44.255   
 44.664  44.843    44.159  45.406    43.705  45.956    43.319  46.457   
 42.925  46.997    42.598  47.471    42.257  47.986    41.942  48.484   
 41.647  48.967    41.340  49.488    41.055  49.988    40.790  50.470   
 40.513  50.985    40.226  51.538    39.970  52.043    39.721  52.548   
 39.462  53.085    39.193  53.658    38.928  54.238    38.694  54.759   
 38.452  55.311    38.200  55.898    37.937  56.523    37.669  57.177   
 37.441  57.742    37.205  58.339    36.959  58.972    36.704  59.643   
 36.437  60.355    36.175  61.072    35.944  61.714    35.705  62.390   
 35.456  63.106    35.197  63.863    34.927  64.666    34.646  65.519   
] scurve 
 62.000  56.400 moveto 
[  62.000  56.400    61.680  55.641    61.548  55.333    61.427  55.055   
 61.317  54.805    61.216  54.578    61.124  54.372    61.039  54.185   
 60.962  54.014    60.891  53.859    60.825  53.717    60.765  53.586   
 60.709  53.467    60.658  53.357    60.610  53.256    60.567  53.164   
 60.526  53.078    60.489  53.000    60.454  52.927    60.422  52.860   
 60.392  52.798    60.364  52.741    60.339  52.688    60.315  52.639   
 60.293  52.594    60.273  52.552    60.254  52.513    60.236  52.477   
 60.220  52.443    60.205  52.412    60.190  52.384    60.177  52.357   
 60.165  52.332    60.154  52.309    60.143  52.288    60.133  52.268   
 60.124  52.249    60.116  52.232    60.108  52.216    60.100  52.201   
 60.093  52.187    60.087  52.174    60.081  52.162    60.076  52.151   
 60.070  52.141    60.066  52.131    60.061  52.122   ] scurve 
 62.000  56.400 moveto 
[  62.000  56.400    62.383  57.332    62.559  57.770    62.752  58.256   
 62.962  58.797    63.190  59.390    63.391  59.926    63.610  60.516   
 63.848  61.168    64.107  61.891    64.364  62.621    64.609  63.328   
 64.873  64.104    65.159  64.961    65.469  65.910   ] scurve 
 45.716  64.850 moveto 
[  45.716  64.850    45.886  63.859    45.968  63.411    46.049  62.980   
 46.130  62.565    46.211  62.166    46.292  61.782    46.372  61.412   
 46.452  61.055    46.532  60.710    46.611  60.378    46.691  60.057   
 46.770  59.747    46.849  59.447    46.928  59.158    47.006  58.878   
 47.085  58.607    47.163  58.344    47.241  58.091    47.320  57.845   
 47.397  57.607    47.475  57.376    47.553  57.152    47.631  56.935   
 47.708  56.725    47.785  56.521    47.863  56.324    47.940  56.132   
 48.017  55.946    48.094  55.765    48.171  55.589    48.248  55.419   
 48.325  55.254    48.402  55.093    48.479  54.937    48.555  54.786   
 48.632  54.639    48.709  54.496    48.785  54.357    48.862  54.222   
 48.939  54.091    49.015  53.963    49.092  53.840    49.168  53.719   
 49.245  53.603    49.322  53.489    49.398  53.379    49.475  53.272   
 49.551  53.168    49.628  53.067    49.704  52.969    49.781  52.873   
 49.857  52.781    49.934  52.691    50.010  52.604    50.087  52.519   
 50.163  52.437    50.240  52.357    50.316  52.280    50.393  52.205   
 50.469  52.132    50.546  52.062    50.622  51.994    50.699  51.927   
 50.775  51.863    50.852  51.801    50.928  51.741    51.005  51.683   
 51.081  51.627    51.157  51.573    51.234  51.520    51.310  51.469   
 51.386  51.420    51.463  51.373    51.539  51.328    51.615  51.284   
 51.691  51.241    51.767  51.201    51.844  51.162    51.920  51.124   
 51.995  51.088    52.071  51.053    52.147  51.020    52.223  50.988   
 52.298  50.957    52.374  50.928    52.449  50.900    52.525  50.874   
 52.600  50.848    52.675  50.824    52.750  50.801    52.825  50.780   
 52.900  50.759    52.974  50.740    53.049  50.721    53.123  50.704   
 53.197  50.688    53.271  50.673    53.345  50.659    53.418  50.646   
 53.492  50.633    53.565  50.622    53.638  50.612    53.710  50.603   
 53.783  50.594    53.855  50.586    53.927  50.580    53.999  50.574   
 54.070  50.569    54.142  50.564    54.213  50.561    54.283  50.558   
 54.354  50.555    54.424  50.554    54.493  50.553    54.563  50.553   
 54.632  50.554    54.701  50.555    54.769  50.557    54.837  50.559   
 54.905  50.562    54.972  50.565    55.039  50.569    55.105  50.574   
 55.171  50.579    55.237  50.584    55.303  50.590    55.367  50.597   
 55.432  50.604    55.496  50.611    55.559  50.618    55.623  50.626   
 55.685  50.635    55.748  50.644    55.809  50.653    55.871  50.662   
 55.931  50.672    55.992  50.682    56.051  50.692    56.111  50.703   
 56.170  50.714    56.228  50.725    56.287  50.736    56.358  50.750   
 56.429  50.765    56.499  50.780    56.568  50.795    56.636  50.811   
 56.703  50.826    56.770  50.842    56.835  50.858    56.900  50.875   
 56.964  50.891    57.026  50.907    57.088  50.924    57.150  50.940   
 57.210  50.957    57.269  50.974    57.328  50.990    57.385  51.007   
 57.442  51.024    57.498  51.040    57.553  51.057    57.607  51.073   
 57.660  51.090    57.712  51.106    57.763  51.123    57.814  51.139   
 57.863  51.155    57.912  51.171    57.960  51.187    58.007  51.203   
 58.053  51.218    58.098  51.234    58.143  51.249    58.186  51.265   
 58.229  51.280    58.271  51.294    58.312  51.309    58.352  51.324   
 58.392  51.338    58.431  51.352    58.468  51.366    58.506  51.380   
 58.542  51.393    58.578  51.407    58.613  51.420    58.647  51.433   
 58.680  51.445    58.713  51.458   ] scurve 
[  58.713  51.458    58.745  51.470    58.776  51.482    58.807  51.494   
 58.837  51.506    58.866  51.518    58.895  51.529    58.923  51.540   
 58.950  51.551    58.977  51.561    59.003  51.572    59.029  51.582   
 59.054  51.592    59.078  51.602    59.102  51.612    59.125  51.621   
 59.148  51.631    59.170  51.640    59.192  51.649    59.213  51.657   
 59.233  51.666    59.253  51.674    59.273  51.682    59.292  51.690   
 59.311  51.698    59.329  51.706    59.347  51.713    59.364  51.721   
 59.381  51.728    59.398  51.735    59.414  51.742    59.429  51.748   
 59.445  51.755    59.459  51.761   ] scurve 
 45.716  64.850 moveto 
[  45.716  64.850    45.543  65.927    45.460  66.481   ] scurve 
 55.713  60.566 moveto 
[  55.713  60.566    55.961  59.713    56.039  59.462    56.116  59.219   
 56.193  58.986    56.268  58.761    56.343  58.544    56.417  58.335   
 56.490  58.134    56.562  57.939    56.633  57.752    56.703  57.571   
 56.772  57.396    56.840  57.227    56.907  57.064    56.974  56.907   
 57.039  56.755    57.103  56.608    57.167  56.466    57.229  56.328   
 57.291  56.195    57.351  56.066    57.411  55.942    57.469  55.822   
 57.527  55.705    57.584  55.592    57.639  55.483    57.694  55.378   
 57.748  55.275    57.801  55.176    57.853  55.080    57.904  54.987   
 57.953  54.897    58.003  54.810    58.051  54.726    58.098  54.644   
 58.144  54.565    58.189  54.488    58.234  54.413    58.277  54.341   
 58.320  54.271    58.362  54.203    58.403  54.137    58.443  54.074   
 58.482  54.012    58.521  53.952    58.558  53.894    58.595  53.837   
 58.631  53.783    58.666  53.730    58.701  53.678    58.734  53.628   
 58.767  53.580    58.799  53.533    58.831  53.488    58.861  53.443   
 58.891  53.401    58.920  53.359    58.949  53.319    58.977  53.280   
 59.004  53.242    59.031  53.205    59.057  53.169    59.082  53.135   
 59.107  53.101    59.131  53.069    59.154  53.037    59.177  53.006   
 59.199  52.977    59.221  52.948    59.242  52.920    59.263  52.893   
 59.283  52.866    59.303  52.841    59.322  52.816    59.341  52.792   
 59.359  52.768    59.377  52.746    59.394  52.724    59.411  52.702   
 59.427  52.682    59.443  52.661    59.459  52.642    59.474  52.623   
 59.489  52.605    59.503  52.587    59.517  52.569    59.530  52.553   
 59.544  52.536    59.556  52.521    59.569  52.505    59.581  52.490   
 59.593  52.476    59.604  52.462    59.616  52.448    59.627  52.435   
 59.637  52.422    59.647  52.410    59.658  52.398    59.667  52.386   
 59.677  52.374    59.686  52.363    59.695  52.353    59.704  52.342   
 59.712  52.332    59.720  52.322    59.728  52.313    59.736  52.304   
 59.744  52.295    59.751  52.286    59.758  52.278    59.765  52.269   
 59.772  52.261    59.779  52.254    59.785  52.246    59.791  52.239   
] scurve 
 55.713  60.566 moveto 
[  55.713  60.566    55.456  61.524    55.373  61.860    55.288  62.210   
 55.203  62.576    55.120  62.943    55.051  63.260    54.981  63.589   
 54.911  63.929    54.841  64.283    54.770  64.649    54.699  65.030   
 54.627  65.426   ] scurve 
  8.584  61.994 moveto 
[   8.584  61.994     9.893  61.145    11.062  60.384    12.136  59.684   
 13.127  59.036    14.046  58.434    14.900  57.872    15.698  57.347   
 16.444  56.853    17.145  56.389    17.805  55.950    18.428  55.535   
 19.017  55.142    19.575  54.768    20.105  54.413    20.609  54.073   
 21.089  53.749    21.548  53.439    21.986  53.142    22.405  52.857   
 22.807  52.584    23.193  52.321    23.563  52.067    23.919  51.823   
 24.262  51.587    24.592  51.359    24.911  51.139    25.219  50.926   
 25.516  50.720    25.803  50.520    26.082  50.326    26.351  50.138   
 26.612  49.955    26.865  49.778    27.111  49.605    27.350  49.437   
 27.582  49.273    27.807  49.113    28.027  48.958    28.240  48.806   
 28.448  48.658    28.651  48.513    28.884  48.347    29.123  48.175   
 29.355  48.008    29.580  47.846    29.798  47.688    30.011  47.535   
 30.217  47.385    30.418  47.238    30.614  47.095    30.804  46.956   
 31.032  46.789    31.256  46.624    31.474  46.464    31.685  46.307   
 31.890  46.155    32.089  46.007    32.283  45.862    32.472  45.721   
 32.655  45.583    32.878  45.415    33.094  45.252    33.303  45.093   
 33.506  44.939    33.702  44.788    33.894  44.642    34.080  44.498   
 34.277  44.346    34.495  44.177    34.707  44.012    34.912  43.852   
 35.111  43.695    35.304  43.543    35.491  43.395    35.713  43.218   
 35.932  43.043    36.144  42.873    36.349  42.708    36.548  42.547   
 36.774  42.362    37.006  42.172    37.229  41.988    37.444  41.809   
 37.690  41.603    37.940  41.393    38.180  41.189    38.447  40.960   
 38.724  40.722    39.012  40.471    39.327  40.193    39.688  39.872   
 40.097  39.501    40.627  39.012    41.348  38.328    42.174  37.514   
 42.923  36.742    43.615  35.996    44.159  35.383    44.595  34.873   
 44.982  34.404    45.312  33.992    45.609  33.609    45.901  33.222   
 46.157  32.870    46.386  32.550    46.612  32.223    46.837  31.890   
 47.060  31.549    47.254  31.245    47.431  30.960    47.607  30.669   
 47.783  30.371    47.958  30.065    48.133  29.752    48.307  29.430   
 48.475  29.111    48.615  28.839    48.754  28.560    48.893  28.274   
 49.032  27.980    49.171  27.678    49.310  27.368    49.449  27.048   
 49.588  26.718    49.727  26.377    49.861  26.038    49.972  25.748   
 50.083  25.450    50.194  25.144    50.305  24.828    50.416  24.503   
 50.527  24.167    50.638  23.821    50.749  23.463    50.861  23.093   
 50.972  22.710    51.083  22.314    51.185  21.933    51.274  21.594   
 51.363  21.244    51.451  20.883    51.540  20.511    51.629  20.127   
 51.717  19.729    51.805  19.318    51.894  18.892    51.982  18.451   
 52.070  17.993    52.158  17.519    52.246  17.025    52.321  16.588   
 52.391  16.165    52.462  15.728    52.532  15.276    52.601  14.809   
 52.671  14.325    52.741  13.824    52.811  13.305    52.880  12.766   
 52.949  12.206    53.019  11.624    53.088  11.019    53.157  10.389   
 53.212   9.866    53.267   9.326    53.322   8.766    53.376   8.187   
 53.431   7.586    53.485   6.964    53.540   6.318    53.594   5.647   
 53.648   4.950    53.702   4.224   ] scurve 
  8.584  61.994 moveto 
[   8.584  61.994     7.139  62.925     5.751  63.817     4.383  64.693   
] scurve 
grestore}
\LabelTeX   98.000  38.400 $x$\ELTX
\LabelTeX   46.200  68.000 $y$\ELTX
\LabelTeX   57.600  38.400 $1$\ELTX
\LabelTeX   46.700  50.500 $1$\ELTX
\EndFig
\vskip4mm

\subsection 1.6. \'Equations homog\`enes|

\noindent Une \'equation homog\`ene est une \'equation qui peut se
mettre sous la forme
$$
y'=f\ \Big(\frac{y}{x}\Big)\quad\hbox{\rm o\`u}\quad f:I\rightarrow\bR
\quad\hbox{est continue.}\leqno{\rm (E)}
$$
C'est le cas par exemple des \'equations $y'=\dfrac{P(x,y)}{Q(x,y)}$
o\`u $P,Q$ sont des polyn\^omes homog\`enes de m\^eme degr\'e $d$~: une
division par $x^d$ au num\'erateur et au d\'enominateur nous ram\`ene
\`a $y'=\dfrac{P(1,y/x)}{Q(1,y/x)}$.

\bigskip
\state M\'ethode -- |  On pose $z=\dfrac{y}{x}$, c'est-\`a-dire $y=xz$.
Il vient
$$
y'=z+xz'=f(z),
$$
donc $z$ satisfait l'\'equation \`a variables s\'epar\'ees
$$
z'=\frac{f(z)-z}{x}.
$$
{\parindent=0.8pc
\item{$\bullet$} On a d'une part les solutions singuli\`eres
$$
z(x)=z_j,\qquad y(x)=z_jx\qquad\hbox{(droites passant par $0$),}
$$
o\`u $\{z_j\}$ est l'ensemble des racines de $f(z)=z$.\par}
\medskip

{\parindent=0.8pc
\item{$\bullet$} Pour $f(z)\not=z$ on peut \'ecrire
$$
\eqalign{
&\frac{dz}{f(z)-z}=\frac{dx}{x},\cr
&F(z)=\ln\,|x|+C=\ln\,(\lambda x),\qquad\lambda\in\bR^*,\cr
}$$
o\`u $F$ est une primitive de $z\mapsto 1/(f(z)-z)$ sur $]z_j,z_{j+1}[$. On en d\'eduit
que\break 
$z=F^{-1}(\ln\,(\lambda x))$, d'o\`u la famille de courbes int\'egrales
$$
C_{\lambda}: y=xF^{-1}(\ln\,(\lambda x)),
$$
d\'efinies dans le secteur angulaire $z_j<\dfrac{y}{x}<z_{j+1}$, $\lambda x>0$.
\par}

En cas de divergence de $F$ aux points $z_j,z_{j+1}$, on a
$F^{-1}:\,]-\infty,+\infty[\rightarrow ]z_{j},z_{j+1}[$ monotone
bijective et $\dfrac{y}{x}\rightarrow z_j$ ou $z_{j+1}$ quand
$x\rightarrow 0$ ou $\infty$. On a donc d'une part une branche infinie
de direction asymptotique $y=z_{j+1}x$ (resp. $y=z_jx$) et une tangente
$y=z_jx$ (resp. $y=z_{j+1}x$) au point $0$ si $F$ est croissante (resp.
d\'ecroissante). Noter que la droite $y=z_j x$ n'est pas
n\'ecessairement asymptote~: voir l'exemple ci-dessous.

Observons enfin que les lignes isoclines sont les droites $y=mx$, la
pente correspondante \'etant $f(m)$. Le champ des tangentes est donc
invariant par les homoth\'eties de centre $O$. Ceci permet de voir que
{\it l'homoth\'etique d'une courbe int\'egrale est encore une courbe
int\'egrale.}

\bigskip
\majedeb
\state Exercice -- |  {\it V\'erifier que
$C_{\lambda}=h_{1/\lambda}(C_1)$ o\`u $h_{\lambda}(x,y)=(\lambda
x,\lambda y)$.}  \par\majefin\par

\InsertFig 16.000 84.000 {
1 mm unit  /nbpt 10 def
 79.000  13.000 moveto 
-30.000 rotmem 
  1.000 nbpt mul it  (f) show 0.5 0 rmoveto currentpoint 
  1.000 nbpt mul rm  <28> show currentpoint 
  1.000 nbpt mul it  (m) show currentpoint 
  1.000 nbpt mul rm  <29> show ( = ) show currentpoint 
  1.000 nbpt mul it  (m) show currentpoint 
clear 
 16.500   3.700 moveto 
  1.000 nbpt mul it  (y = z) show currentpoint 
  0.000  -0.175 nbpt mul rmoveto 
  0.700 nbpt mul rm  (1) show currentpoint 
1 index 3 index moveto 
  1.000 nbpt mul it  (x) show currentpoint
clear 
rotrev 
 94.000  58.000 moveto 
 30.000 rotmem 
  3.500   1.000 moveto 
  1.000 nbpt mul it  (y = z) show currentpoint 
  0.000  -0.175 nbpt mul rmoveto 
  0.700 nbpt mul rm  (2) show currentpoint 
1 index 3 index moveto 
  1.000 nbpt mul it  (x) show currentpoint 
clear
rotrev 
  0.000  35.000 moveto 
100.000   0.000 vector 
 50.000   0.000 moveto 
 70.000  90.000 vector 
[   6.000   1.500   0.750   1.500 ] 0 setdash
 50.000  35.000 moveto
 29.793   0.000 lineto
stroke
 50.000  35.000 moveto
 70.207  70.000 lineto
stroke
[ ] 0 setdash
 55.850  45.100 moveto 
  4.000 -16.000 vector 
  4.000 164.000 vector 
 61.250  54.490 moveto 
  5.000 -16.000 vector 
  5.000 164.000 vector 
 66.450  63.500 moveto 
  6.000 -16.000 vector 
  6.000 164.000 vector
  0.452 setlinewidth
 85.000  14.793 moveto
  0.700 disk
  4.000 -30.000 vector
  4.000 150.000 vector
  0.113 setlinewidth
gsave 
  5.000   5.000 moveto 
 90.000  60.000 rect  clip  newpath
  0.000   6.132 moveto 
100.000  63.868 lineto 
stroke 
  0.000  63.868 moveto 
100.000   6.132 lineto 
stroke 
 65.000  35.000 moveto 
[  65.000  35.000    64.082  35.862    63.365  36.458    62.886  36.818   
 62.513  37.078    62.173  37.299    61.908  37.461    61.650  37.609   
 61.399  37.746    61.199  37.848    61.008  37.941    60.822  38.026   
 60.640  38.105    60.464  38.177    60.292  38.243    60.147  38.296   
 60.016  38.341    59.889  38.382    59.764  38.420    59.642  38.455   
 59.523  38.487    59.407  38.516    59.294  38.542    59.184  38.566   
 59.076  38.587    58.970  38.606    58.868  38.623    58.767  38.637   
 58.669  38.650    58.574  38.661    58.480  38.669    58.389  38.677   
 58.307  38.682    58.237  38.685    58.168  38.688    58.101  38.689   
 58.035  38.690    57.970  38.690    57.906  38.689    57.843  38.688   
 57.782  38.685    57.721  38.682    57.662  38.679    57.604  38.675   
 57.546  38.670    57.490  38.664    57.434  38.659    57.380  38.652   
 57.326  38.645    57.273  38.638    57.221  38.630    57.170  38.622   
 57.119  38.614    57.070  38.605    57.021  38.595    56.973  38.586   
 56.925  38.576    56.878  38.566    56.832  38.555    56.787  38.545   
 56.742  38.534    56.697  38.522    56.654  38.511    56.611  38.499   
 56.568  38.487    56.526  38.475    56.485  38.463    56.444  38.451   
 56.403  38.438    56.363  38.425    56.324  38.412    56.285  38.399   
 56.246  38.386    56.208  38.373    56.171  38.360    56.133  38.346   
 56.097  38.333    56.060  38.319    56.024  38.306    55.988  38.292   
 55.953  38.278    55.918  38.264    55.883  38.250    55.849  38.237   
 55.815  38.223    55.782  38.209    55.748  38.195    55.715  38.180   
 55.683  38.166    55.650  38.152    55.618  38.138    55.586  38.124   
 55.555  38.110    55.524  38.096    55.493  38.082    55.462  38.067   
 55.431  38.053    55.401  38.039    55.371  38.025    55.341  38.011   
 55.312  37.997    55.283  37.983    55.254  37.969    55.225  37.955   
 55.196  37.941    55.168  37.927    55.140  37.913    55.112  37.899   
 55.084  37.885    55.057  37.871    55.029  37.857    55.002  37.843   
 54.975  37.829    54.948  37.816    54.922  37.802    54.895  37.788   
 54.869  37.775    54.843  37.761    54.817  37.748    54.791  37.734   
 54.766  37.721    54.741  37.707    54.715  37.694    54.690  37.681   
 54.665  37.667    54.641  37.654    54.616  37.641    54.592  37.628   
 54.568  37.615    54.543  37.602    54.519  37.589    54.496  37.576   
 54.472  37.563    54.448  37.550    54.425  37.537    54.402  37.525   
 54.379  37.512    54.356  37.499    54.333  37.487    54.310  37.474   
 54.288  37.462    54.265  37.450    54.243  37.437    54.221  37.425   
 54.199  37.413    54.177  37.400    54.155  37.388    54.134  37.376   
 54.112  37.364    54.091  37.352    54.069  37.340    54.048  37.329   
 54.027  37.317    54.006  37.305    53.985  37.293    53.965  37.282   
 53.944  37.270    53.924  37.259    53.903  37.247    53.883  37.236   
 53.863  37.224    53.843  37.213    53.823  37.202    53.803  37.190   
 53.783  37.179    53.764  37.168    53.744  37.157    53.725  37.146   
 53.706  37.135    53.687  37.124    53.667  37.113    53.648  37.103   
 53.630  37.092    53.611  37.081    53.592  37.071    53.574  37.060   
 53.555  37.049    53.537  37.039    53.519  37.029    53.500  37.018   
 53.482  37.008    53.464  36.998    53.446  36.987    53.429  36.977   
 53.411  36.967    53.393  36.957    53.376  36.947    53.359  36.937   
 53.341  36.927    53.324  36.917   ] scurve 
[  53.324  36.917    53.307  36.907    53.290  36.898    53.273  36.888   
 53.256  36.878    53.239  36.869    53.222  36.859    53.206  36.850   
 53.189  36.840    53.173  36.831    53.157  36.821    53.140  36.812   
 53.124  36.803    53.108  36.793    53.092  36.784    53.076  36.775   
 53.060  36.766    53.045  36.757    53.029  36.748    53.013  36.739   
 52.998  36.730    52.982  36.721    52.967  36.712    52.952  36.704   
 52.937  36.695    52.921  36.686    52.906  36.677    52.891  36.669   
 52.877  36.660    52.862  36.652    52.847  36.643    52.832  36.635   
 52.818  36.626    52.803  36.618    52.789  36.610    52.775  36.601   
 52.760  36.593    52.746  36.585    52.732  36.577    52.718  36.569   
 52.704  36.561    52.690  36.553    52.676  36.545    52.662  36.537   
 52.649  36.529    52.635  36.521    52.622  36.513    52.608  36.506   
 52.595  36.498    52.581  36.490    52.568  36.482    52.555  36.475   
 52.542  36.467    52.529  36.460    52.516  36.452    52.503  36.445   
 52.490  36.437    52.477  36.430    52.464  36.423    52.452  36.415   
 52.439  36.408    52.426  36.401    52.414  36.394    52.402  36.386   
 52.389  36.379    52.377  36.372    52.365  36.365    52.353  36.358   
 52.341  36.351    52.328  36.344    52.317  36.337    52.305  36.330   
 52.293  36.324    52.281  36.317    52.269  36.310    52.258  36.303   
 52.246  36.297    52.234  36.290    52.223  36.283    52.212  36.277   
 52.200  36.270    52.189  36.264    52.178  36.257    52.166  36.251   
 52.155  36.244    52.144  36.238    52.133  36.232    52.122  36.225   
 52.111  36.219    52.100  36.213    52.090  36.206    52.079  36.200   
 52.068  36.194    52.058  36.188    52.047  36.182    52.036  36.176   
 52.026  36.170    52.016  36.164    52.005  36.158    51.995  36.152   
 51.985  36.146    51.974  36.140    51.964  36.134    51.954  36.128   
 51.944  36.122    51.934  36.117    51.924  36.111    51.914  36.105   
 51.905  36.100   ] scurve 
 65.000  35.000 moveto 
[  65.000  35.000    65.918  34.026    66.637  33.185    67.100  32.610   
 67.455  32.151    67.773  31.730    68.048  31.356    68.319  30.982   
 68.532  30.682    68.744  30.383    68.952  30.084    69.159  29.788   
 69.337  29.530    69.499  29.295    69.660  29.062    69.819  28.830   
 69.977  28.601    70.135  28.373    70.291  28.147    70.446  27.923   
 70.592  27.714    70.715  27.539    70.838  27.365    70.960  27.192   
 71.082  27.021    71.204  26.852    71.325  26.684    71.447  26.518   
 71.568  26.354    71.689  26.191    71.809  26.029    71.930  25.870   
 72.051  25.712    72.172  25.555    72.292  25.400    72.413  25.247   
 72.534  25.095    72.655  24.945    72.777  24.796    72.898  24.649   
 72.998  24.530    73.095  24.414    73.193  24.300    73.291  24.186   
 73.389  24.073    73.487  23.961    73.586  23.849    73.685  23.739   
 73.784  23.630    73.883  23.521    73.983  23.413    74.082  23.306   
 74.183  23.200    74.283  23.094    74.384  22.989    74.485  22.885   
 74.586  22.782    74.688  22.679    74.789  22.577    74.892  22.476   
 74.994  22.375    75.097  22.275    75.201  22.176    75.305  22.077   
 75.409  21.978    75.513  21.881    75.618  21.784    75.723  21.687   
 75.829  21.591    75.935  21.495    76.041  21.400    76.148  21.306   
 76.256  21.212    76.363  21.118    76.471  21.025    76.580  20.932   
 76.689  20.840    76.798  20.748    76.908  20.656    77.019  20.565   
 77.129  20.474    77.240  20.383    77.352  20.293    77.464  20.203   
 77.577  20.113    77.690  20.024    77.803  19.934    77.917  19.846   
 78.032  19.757    78.147  19.668    78.262  19.580    78.378  19.492   
 78.494  19.405    78.611  19.317    78.728  19.229    78.846  19.142   
 78.964  19.055    79.083  18.968    79.203  18.881    79.322  18.795   
 79.443  18.708    79.563  18.621    79.685  18.535    79.807  18.449   
 79.909  18.377    80.007  18.308    80.106  18.239    80.204  18.170   
 80.304  18.101    80.403  18.032    80.503  17.963    80.603  17.894   
 80.704  17.826    80.805  17.757    80.906  17.688    81.007  17.619   
 81.109  17.550    81.211  17.482    81.314  17.413    81.417  17.344   
 81.520  17.275    81.624  17.206    81.728  17.137    81.832  17.068   
 81.937  16.999    82.042  16.930    82.147  16.861    82.253  16.792   
 82.359  16.723    82.465  16.653    82.572  16.584    82.679  16.515   
 82.787  16.445    82.895  16.376    83.003  16.306    83.111  16.236   
 83.220  16.167    83.330  16.097    83.439  16.027    83.550  15.957   
 83.660  15.887    83.771  15.817    83.882  15.746    83.994  15.676   
 84.105  15.605    84.218  15.535    84.330  15.464    84.443  15.393   
 84.557  15.322    84.671  15.251    84.785  15.180    84.900  15.109   
 85.015  15.037    85.130  14.966    85.246  14.894    85.362  14.822   
 85.478  14.750    85.595  14.678    85.712  14.606    85.830  14.534   
 85.948  14.461    86.067  14.389    86.186  14.316    86.305  14.243   
 86.424  14.170    86.545  14.097    86.665  14.023    86.786  13.950   
 86.907  13.876    87.029  13.802    87.151  13.728    87.273  13.654   
 87.396  13.579    87.520  13.505    87.643  13.430    87.767  13.355   
 87.892  13.280    88.017  13.205    88.142  13.129    88.268  13.054   
 88.394  12.978    88.521  12.902    88.648  12.826    88.776  12.749   
 88.903  12.673    89.032  12.596    89.161  12.519    89.290  12.442   
 89.419  12.365    89.549  12.287   ] scurve 
[  89.549  12.287    89.680  12.209    89.811  12.131    89.942  12.053   
 90.074  11.975    90.206  11.896    90.339  11.817    90.472  11.738   
 90.605  11.659    90.739  11.580    90.874  11.500    91.009  11.420   
 91.144  11.340    91.280  11.260    91.416  11.179    91.553  11.098   
 91.690  11.017    91.828  10.936    91.966  10.855    92.104  10.773   
 92.243  10.691    92.382  10.609    92.522  10.527    92.663  10.444   
 92.804  10.361    92.945  10.278    93.087  10.195    93.229  10.111   
 93.372  10.027    93.515   9.943    93.658   9.859    93.802   9.774   
 93.947   9.689    94.092   9.604    94.238   9.519    94.384   9.433   
 94.530   9.348    94.677   9.262    94.825   9.175    94.973   9.089   
 95.121   9.002    95.270   8.915   ] scurve 
 80.000  35.000 moveto 
[  80.000  35.000    78.621  36.316    77.284  37.472    76.334  38.221   
 75.590  38.766    74.938  39.215    74.417  39.553    73.918  39.861   
 73.431  40.145    73.051  40.356    72.678  40.554    72.313  40.738   
 71.957  40.909    71.609  41.068    71.295  41.204    71.029  41.314   
 70.768  41.417    70.513  41.513    70.264  41.603    70.019  41.686   
 69.780  41.764    69.547  41.835    69.318  41.901    69.094  41.962   
 68.875  42.017    68.662  42.068    68.452  42.114    68.248  42.156   
 68.055  42.192    67.899  42.220    67.745  42.244    67.594  42.267   
 67.445  42.287    67.300  42.305    67.156  42.320    67.016  42.334   
 66.878  42.346    66.742  42.356    66.608  42.364    66.477  42.370   
 66.349  42.375    66.222  42.378    66.098  42.380    65.976  42.380   
 65.855  42.379    65.737  42.377    65.621  42.373    65.507  42.368   
 65.394  42.362    65.283  42.355    65.174  42.347    65.067  42.338   
 64.962  42.327    64.858  42.316    64.756  42.304    64.655  42.291   
 64.556  42.278    64.458  42.263    64.362  42.248    64.267  42.232   
 64.173  42.216    64.081  42.199    63.990  42.181    63.900  42.163   
 63.812  42.144    63.724  42.125    63.638  42.105    63.553  42.084   
 63.469  42.064    63.387  42.043    63.305  42.021    63.224  41.999   
 63.144  41.977    63.065  41.954    62.987  41.931    62.910  41.908   
 62.834  41.885    62.759  41.861    62.685  41.837    62.611  41.813   
 62.538  41.788    62.466  41.764    62.395  41.739    62.325  41.714   
 62.255  41.689    62.186  41.663    62.118  41.638    62.050  41.612   
 61.983  41.587    61.917  41.561    61.851  41.535    61.786  41.509   
 61.722  41.483    61.658  41.457    61.594  41.430    61.532  41.404   
 61.470  41.378    61.408  41.351    61.347  41.325    61.286  41.298   
 61.226  41.272    61.167  41.245    61.108  41.219    61.049  41.192   
 60.991  41.166    60.933  41.139    60.876  41.113    60.819  41.086   
 60.763  41.060    60.707  41.033    60.651  41.007    60.596  40.980   
 60.542  40.954    60.487  40.928    60.434  40.901    60.380  40.875   
 60.327  40.849    60.274  40.823    60.222  40.797    60.170  40.771   
 60.118  40.745    60.067  40.719    60.016  40.693    59.965  40.667   
 59.915  40.641    59.865  40.615    59.815  40.590    59.766  40.564   
 59.717  40.539    59.668  40.513    59.620  40.488    59.572  40.463   
 59.524  40.437    59.476  40.412    59.429  40.387    59.382  40.362   
 59.336  40.337    59.289  40.313    59.243  40.288    59.197  40.263   
 59.152  40.239    59.106  40.214    59.061  40.190    59.017  40.166   
 58.972  40.141    58.928  40.117    58.884  40.093    58.840  40.069   
 58.796  40.046    58.753  40.022    58.710  39.998    58.667  39.975   
 58.625  39.951    58.582  39.928    58.540  39.905    58.499  39.881   
 58.457  39.858    58.416  39.835    58.374  39.812    58.333  39.790   
 58.293  39.767    58.252  39.744    58.212  39.722    58.172  39.699   
 58.132  39.677    58.092  39.655    58.053  39.633    58.013  39.611   
 57.974  39.589    57.936  39.567    57.897  39.545    57.859  39.524   
 57.820  39.502    57.782  39.481    57.744  39.459    57.707  39.438   
 57.669  39.417    57.632  39.396    57.595  39.375    57.558  39.354   
 57.521  39.333    57.485  39.312    57.449  39.292    57.412  39.271   
 57.377  39.251    57.341  39.230    57.305  39.210    57.270  39.190   
 57.235  39.170    57.199  39.150   ] scurve 
[  57.199  39.150    57.165  39.130    57.130  39.110    57.095  39.091   
 57.061  39.071    57.027  39.051    56.993  39.032    56.959  39.013   
 56.925  38.993    56.892  38.974    56.858  38.955    56.825  38.936   
 56.792  38.917    56.759  38.899    56.727  38.880    56.694  38.861   
 56.662  38.843    56.630  38.824    56.598  38.806    56.566  38.788   
 56.534  38.769    56.502  38.751    56.471  38.733    56.440  38.715   
 56.409  38.697    56.378  38.680    56.347  38.662    56.316  38.644   
 56.286  38.627    56.255  38.609    56.225  38.592    56.195  38.575   
 56.165  38.558    56.135  38.540    56.106  38.523    56.076  38.507   
 56.047  38.490    56.018  38.473    55.989  38.456    55.960  38.440   
 55.931  38.423    55.902  38.407    55.874  38.390    55.846  38.374   
 55.817  38.358    55.789  38.341    55.761  38.325    55.734  38.309   
 55.706  38.293    55.678  38.278    55.651  38.262    55.624  38.246   
 55.597  38.230    55.570  38.215    55.543  38.199    55.516  38.184   
 55.490  38.169    55.463  38.153    55.437  38.138    55.410  38.123   
 55.384  38.108    55.358  38.093    55.333  38.078    55.307  38.063   
 55.281  38.049    55.256  38.034    55.230  38.019    55.205  38.005   
] scurve 
 80.000  35.000 moveto 
[  80.000  35.000    81.379  33.557    82.694  32.065    83.630  30.935   
 84.320  30.066    84.913  29.299    85.438  28.603    85.908  27.967   
 86.318  27.406    86.722  26.844    87.123  26.282    87.447  25.823   
 87.760  25.375    88.072  24.929    88.380  24.484    88.686  24.042   
 88.990  23.601    89.253  23.219    89.493  22.871    89.731  22.525   
 89.968  22.181    90.204  21.839    90.439  21.499    90.673  21.162   
 90.906  20.827    91.138  20.495    91.369  20.165    91.599  19.838   
 91.828  19.513    92.020  19.244    92.203  18.988    92.385  18.734   
 92.567  18.482    92.749  18.232    92.931  17.983    93.113  17.737   
 93.294  17.492    93.475  17.249    93.656  17.007    93.838  16.768   
 94.019  16.530    94.200  16.295    94.381  16.061    94.562  15.828   
 94.743  15.598    94.925  15.369    95.106  15.142    95.288  14.917   
] scurve 
 95.000  35.000 moveto 
[  95.000  35.000    93.344  36.596    91.701  38.058    90.231  39.267   
 89.124  40.114    88.183  40.792    87.402  41.326    86.637  41.823   
 86.024  42.203    85.435  42.552    84.858  42.879    84.293  43.186   
 83.800  43.442    83.365  43.658    82.939  43.862    82.521  44.054   
 82.110  44.235    81.708  44.404    81.314  44.563    80.928  44.712   
 80.550  44.850    80.187  44.977    79.897  45.073    79.612  45.164   
 79.332  45.249    79.056  45.330    78.786  45.404    78.520  45.474   
 78.259  45.540    78.003  45.600    77.752  45.656    77.504  45.708   
 77.262  45.756    77.023  45.799    76.789  45.839    76.560  45.875   
 76.334  45.908    76.112  45.937    75.895  45.963    75.681  45.986   
 75.471  46.006    75.264  46.023    75.062  46.037    74.863  46.049   
 74.667  46.058    74.475  46.065    74.286  46.069    74.100  46.071   
 73.917  46.071    73.738  46.069    73.561  46.064    73.387  46.058   
 73.217  46.051    73.049  46.041    72.883  46.030    72.721  46.017   
 72.560  46.003    72.403  45.987    72.248  45.970    72.095  45.952   
 71.944  45.932    71.796  45.912    71.650  45.890    71.506  45.867   
 71.364  45.843    71.224  45.818    71.086  45.792    70.950  45.765   
 70.816  45.737    70.684  45.709    70.554  45.680    70.425  45.650   
 70.302  45.620    70.202  45.595    70.102  45.570    70.004  45.544   
 69.906  45.518    69.810  45.492    69.714  45.465    69.619  45.438   
 69.525  45.411    69.433  45.383    69.341  45.355    69.249  45.327   
 69.159  45.298    69.069  45.270    68.981  45.241    68.893  45.212   
 68.806  45.182    68.719  45.153    68.633  45.123    68.548  45.093   
 68.464  45.063    68.381  45.033    68.298  45.002    68.216  44.972   
 68.134  44.941    68.053  44.911    67.973  44.880    67.893  44.849   
 67.814  44.818    67.736  44.786    67.658  44.755    67.581  44.724   
 67.504  44.692    67.428  44.661    67.352  44.629    67.277  44.598   
 67.203  44.566    67.129  44.535    67.055  44.503    66.982  44.471   
 66.910  44.439    66.838  44.408    66.766  44.376    66.695  44.344   
 66.625  44.312    66.554  44.280    66.485  44.248    66.416  44.217   
 66.347  44.185    66.278  44.153    66.210  44.121    66.143  44.089   
 66.076  44.057    66.009  44.026    65.943  43.994    65.877  43.962   
 65.811  43.931    65.746  43.899    65.681  43.867    65.617  43.836   
 65.553  43.804    65.489  43.773    65.426  43.741    65.363  43.710   
 65.300  43.679    65.238  43.647    65.176  43.616    65.114  43.585   
 65.053  43.554    64.992  43.523    64.931  43.492    64.871  43.461   
 64.811  43.430    64.751  43.399    64.692  43.369    64.633  43.338   
 64.574  43.307    64.516  43.277    64.457  43.246    64.400  43.216   
 64.342  43.186    64.285  43.155    64.228  43.125    64.171  43.095   
 64.114  43.065    64.058  43.035    64.002  43.006    63.946  42.976   
 63.891  42.946    63.836  42.916    63.781  42.887    63.726  42.858   
 63.672  42.828    63.618  42.799    63.564  42.770    63.510  42.741   
 63.457  42.712    63.404  42.683    63.351  42.654    63.298  42.625   
 63.246  42.596    63.194  42.568    63.142  42.539    63.090  42.511   
 63.038  42.483    62.987  42.454    62.936  42.426    62.885  42.398   
 62.835  42.370    62.784  42.342    62.734  42.315    62.684  42.287   
 62.635  42.259    62.585  42.232    62.536  42.205    62.487  42.177   
 62.438  42.150    62.389  42.123   ] scurve 
[  62.389  42.123    62.341  42.096    62.293  42.069    62.245  42.042   
 62.197  42.015    62.149  41.989    62.102  41.962    62.055  41.935   
 62.008  41.909    61.961  41.883    61.914  41.857    61.868  41.830   
 61.821  41.804    61.775  41.778    61.730  41.753    61.684  41.727   
 61.638  41.701    61.593  41.676    61.548  41.650    61.503  41.625   
 61.458  41.599    61.414  41.574    61.369  41.549    61.325  41.524   
 61.281  41.499    61.237  41.474    61.194  41.449    61.150  41.425   
 61.107  41.400    61.064  41.376    61.021  41.351    60.978  41.327   
 60.936  41.303    60.893  41.278    60.851  41.254    60.809  41.230   
 60.767  41.207    60.725  41.183    60.684  41.159    60.642  41.135   
 60.601  41.112    60.560  41.088    60.519  41.065    60.478  41.042   
 60.438  41.019    60.397  40.995    60.357  40.972    60.317  40.950   
] scurve 
 50.000  25.000 moveto 
[  50.000  25.000    49.151  24.756    48.277  24.583    47.557  24.498   
 47.003  24.468    46.568  24.465    46.144  24.481    45.799  24.507   
 45.479  24.542    45.168  24.587    44.868  24.640    44.599  24.697   
 44.375  24.750    44.159  24.808    43.950  24.870    43.749  24.935   
 43.556  25.002    43.370  25.073    43.192  25.146    43.021  25.220   
 42.858  25.297    42.702  25.374    42.554  25.453    42.412  25.533   
 42.277  25.613    42.149  25.694    42.028  25.775    41.913  25.856   
 41.804  25.937    41.700  26.018    41.603  26.098    41.511  26.178   
 41.425  26.257    41.344  26.336    41.267  26.414    41.196  26.492   
 41.129  26.568    41.067  26.644    41.008  26.719    40.954  26.792   
 40.904  26.865    40.858  26.937    40.815  27.008    40.776  27.077   
 40.740  27.146    40.707  27.214    40.678  27.280    40.651  27.346   
 40.627  27.410    40.606  27.473    40.587  27.536    40.571  27.597   
 40.557  27.657    40.546  27.717    40.536  27.775    40.529  27.832   
 40.524  27.889    40.520  27.944    40.518  27.998    40.518  28.052   
 40.520  28.105    40.523  28.156    40.528  28.207    40.534  28.257   
 40.542  28.307    40.551  28.355    40.561  28.403    40.572  28.450   
 40.584  28.496    40.598  28.541    40.612  28.586    40.628  28.630   
 40.644  28.673    40.662  28.716    40.685  28.768    40.709  28.820   
 40.739  28.879    40.774  28.944    40.820  29.022    40.885  29.123   
 41.046  29.337    41.307  29.625    41.456  29.768    41.575  29.875   
 41.671  29.958    41.768  30.038    41.846  30.100    41.924  30.162   
 42.003  30.222    42.072  30.274    42.135  30.320    42.197  30.366   
 42.260  30.411    42.322  30.455    42.384  30.498    42.446  30.541   
 42.497  30.576    42.546  30.609    42.596  30.642    42.645  30.675   
 42.693  30.708    42.742  30.740    42.791  30.772    42.839  30.803   
 42.887  30.834    42.935  30.865    42.983  30.896    43.030  30.926   
 43.077  30.956    43.124  30.986    43.171  31.015    43.216  31.043   
 43.253  31.066   ] scurve 
 50.000  25.000 moveto 
[  50.000  25.000    50.803  25.305    51.545  25.656    52.102  25.967   
 52.500  26.217    52.808  26.427    53.084  26.629    53.318  26.810   
 53.511  26.967    53.693  27.121    53.864  27.272    54.026  27.418   
 54.172  27.556    54.288  27.667    54.398  27.775    54.502  27.880   
 54.602  27.982    54.698  28.081    54.789  28.176    54.876  28.269   
 54.959  28.358    55.039  28.444    55.116  28.527    55.190  28.607   
 55.260  28.683    55.328  28.757    55.394  28.828    55.457  28.896   
 55.518  28.962    55.577  29.024    55.634  29.084    55.690  29.141   
 55.744  29.196    55.796  29.249    55.847  29.299    55.897  29.347   
 55.946  29.393    55.993  29.436    56.039  29.478    56.085  29.517   
 56.130  29.555    56.174  29.591    56.217  29.624    56.259  29.657   
 56.301  29.687    56.342  29.716    56.383  29.743    56.423  29.769   
 56.463  29.793    56.502  29.816    56.541  29.838    56.580  29.858   
 56.618  29.877    56.656  29.895    56.694  29.911    56.731  29.926   
 56.769  29.941    56.806  29.954    56.843  29.966    56.880  29.977   
 56.916  29.987    56.953  29.996    56.989  30.004    57.026  30.011   
 57.062  30.018    57.099  30.023    57.135  30.028    57.171  30.032   
 57.207  30.035    57.243  30.037    57.280  30.039    57.316  30.040   
 57.352  30.040    57.388  30.040    57.424  30.039    57.461  30.037   
 57.497  30.035    57.533  30.032    57.578  30.027    57.624  30.022   
 57.678  30.014    57.741  30.005    57.821  29.990    57.950  29.961   
 58.308  29.859    58.517  29.786    58.662  29.730    58.785  29.681   
 58.891  29.637    58.991  29.594    59.087  29.552    59.168  29.516   
 59.250  29.478    59.333  29.439    59.417  29.400    59.489  29.366   
 59.557  29.333    59.625  29.300    59.694  29.266    59.763  29.232   
 59.833  29.197    59.903  29.161    59.974  29.126    60.045  29.089   
 60.109  29.057    60.167  29.027    60.225  28.997    60.283  28.966   
 60.342  28.936    60.401  28.905    60.460  28.874    60.520  28.842   
 60.580  28.810    60.640  28.778    60.700  28.746    60.761  28.713   
 60.823  28.680    60.884  28.647    60.946  28.613    61.008  28.580   
 61.071  28.546    61.134  28.511    61.197  28.477    61.261  28.442   
 61.325  28.407    61.390  28.371    61.454  28.336    61.520  28.300   
 61.585  28.264    61.649  28.228    61.702  28.199    61.755  28.169   
 61.808  28.140    61.862  28.110    61.916  28.080    61.970  28.050   
 62.024  28.019    62.079  27.989    62.134  27.958    62.189  27.927   
 62.244  27.896    62.300  27.865    62.356  27.833    62.412  27.802   
 62.468  27.770    62.525  27.738    62.582  27.706    62.639  27.674   
 62.697  27.642    62.754  27.609    62.812  27.576    62.870  27.543   
 62.929  27.510    62.988  27.477    63.047  27.444    63.106  27.410   
 63.165  27.376    63.225  27.342    63.285  27.308    63.346  27.274   
 63.406  27.239    63.467  27.205    63.528  27.170    63.590  27.135   
 63.651  27.100    63.713  27.065    63.776  27.029    63.838  26.993   
 63.901  26.958    63.964  26.922    64.028  26.885    64.091  26.849   
 64.155  26.812    64.220  26.776    64.284  26.739    64.349  26.702   
 64.414  26.664    64.480  26.627    64.545  26.589    64.611  26.551   
 64.678  26.513    64.744  26.475    64.811  26.437    64.879  26.398   
 64.946  26.360    65.014  26.321    65.082  26.282    65.151  26.242   
 65.219  26.203    65.289  26.163   ] scurve 
[  65.289  26.163    65.358  26.123    65.428  26.083    65.498  26.043   
 65.568  26.003    65.639  25.962    65.710  25.921    65.781  25.880   
 65.853  25.839    65.925  25.798    65.997  25.756    66.070  25.715   
 66.143  25.673    66.216  25.631    66.289  25.588    66.363  25.546   
 66.438  25.503    66.512  25.460    66.587  25.417    66.663  25.374   
 66.738  25.330    66.814  25.286    66.891  25.242    66.967  25.198   
 67.044  25.154    67.122  25.109    67.199  25.065    67.277  25.020   
 67.356  24.974    67.435  24.929    67.514  24.883    67.593  24.838   
 67.673  24.792    67.753  24.745    67.834  24.699    67.915  24.652   
 67.996  24.605    68.078  24.558    68.160  24.511    68.242  24.464   
 68.325  24.416    68.408  24.368    68.492  24.320    68.576  24.271   
 68.660  24.223    68.745  24.174    68.830  24.125    68.915  24.075   
 69.001  24.026    69.088  23.976    69.174  23.926    69.261  23.876   
 69.349  23.826    69.437  23.775    69.525  23.724    69.613  23.673   
 69.702  23.622    69.792  23.570    69.882  23.518    69.972  23.466   
 70.063  23.414    70.154  23.361    70.245  23.308    70.337  23.255   
 70.429  23.202    70.522  23.149    70.615  23.095    70.709  23.041   
 70.803  22.987    70.889  22.937    70.964  22.893    71.041  22.850   
 71.117  22.805    71.194  22.761    71.270  22.717    71.348  22.672   
 71.425  22.628    71.503  22.583    71.581  22.538    71.659  22.492   
 71.738  22.447    71.817  22.402    71.896  22.356    71.975  22.310   
 72.055  22.264    72.135  22.218    72.216  22.171    72.296  22.125   
 72.377  22.078    72.458  22.031    72.540  21.984    72.622  21.937   
 72.704  21.890    72.786  21.842    72.869  21.794    72.952  21.746   
 73.035  21.698    73.119  21.650    73.203  21.602    73.287  21.553   
 73.372  21.504    73.456  21.455    73.541  21.406    73.627  21.357   
 73.713  21.307    73.799  21.258    73.885  21.208    73.972  21.158   
 74.059  21.107    74.146  21.057    74.234  21.006    74.322  20.956   
 74.410  20.905    74.499  20.854    74.588  20.802    74.677  20.751   
 74.766  20.699    74.856  20.647    74.946  20.595    75.037  20.543   
 75.128  20.490    75.219  20.438    75.311  20.385    75.403  20.332   
 75.495  20.279    75.587  20.225    75.680  20.171    75.773  20.118   
 75.867  20.064    75.961  20.009    76.055  19.955    76.150  19.900   
 76.244  19.846    76.340  19.791    76.435  19.735    76.531  19.680   
 76.628  19.624    76.724  19.569    76.821  19.513    76.919  19.456   
 77.016  19.400    77.114  19.343    77.213  19.287    77.311  19.230   
 77.411  19.172    77.510  19.115    77.610  19.057    77.710  18.999   
 77.811  18.941    77.912  18.883    78.013  18.824    78.115  18.766   
 78.217  18.707    78.319  18.648    78.422  18.588    78.525  18.529   
 78.629  18.469    78.733  18.409    78.837  18.349    78.941  18.288   
 79.047  18.228    79.152  18.167    79.258  18.106    79.364  18.044   
 79.471  17.983    79.577  17.921    79.685  17.859    79.793  17.797   
 79.901  17.734    80.009  17.672    80.118  17.609    80.227  17.546   
 80.337  17.482    80.447  17.419    80.558  17.355    80.669  17.291   
 80.780  17.227    80.892  17.162    81.004  17.097    81.116  17.032   
 81.229  16.967    81.343  16.902    81.456  16.836    81.571  16.770   
 81.685  16.704    81.800  16.638    81.916  16.571    82.031  16.504   
 82.148  16.437    82.264  16.369   ] scurve 
[  82.264  16.369    82.382  16.302    82.499  16.234    82.617  16.166   
 82.735  16.097    82.854  16.029    82.973  15.960    83.093  15.891   
 83.213  15.822    83.334  15.752    83.455  15.682    83.576  15.612   
 83.698  15.542    83.820  15.471    83.943  15.400    84.066  15.329   
 84.190  15.257    84.314  15.186    84.439  15.114    84.564  15.042   
 84.689  14.969    84.815  14.897    84.941  14.824    85.068  14.750   
 85.195  14.677    85.323  14.603    85.451  14.529    85.580  14.455   
 85.709  14.380    85.839  14.305    85.969  14.230    86.099  14.155   
 86.230  14.079    86.362  14.003    86.494  13.927    86.626  13.850   
 86.759  13.774    86.893  13.697    87.027  13.619    87.161  13.542   
 87.296  13.464    87.431  13.386    87.567  13.307    87.703  13.228   
 87.840  13.149    87.978  13.070    88.116  12.990    88.254  12.910   
 88.393  12.830    88.532  12.750    88.672  12.669    88.812  12.588   
 88.953  12.507    89.079  12.434    89.192  12.368    89.306  12.303   
 89.420  12.237    89.535  12.171    89.650  12.105    89.765  12.038   
 89.880  11.971    89.996  11.905    90.112  11.838    90.228  11.770   
 90.345  11.703    90.462  11.635    90.580  11.567    90.697  11.499   
 90.816  11.431    90.934  11.363    91.053  11.294    91.172  11.225   
 91.291  11.156    91.411  11.087    91.532  11.018    91.652  10.948   
 91.773  10.878    91.894  10.808    92.016  10.738    92.138  10.668   
 92.260  10.597    92.383  10.526    92.506  10.455    92.629  10.384   
 92.753  10.313    92.877  10.241    93.001  10.169    93.126  10.097   
 93.251  10.025    93.377   9.952    93.503   9.879    93.629   9.807   
 93.756   9.733    93.883   9.660    94.010   9.587    94.138   9.513   
 94.266   9.439    94.395   9.365    94.523   9.290    94.653   9.216   
 94.782   9.141    94.912   9.066    95.043   8.990    95.173   8.915   
] scurve 
 50.000  16.000 moveto 
[  50.000  16.000    48.523  15.570    47.002  15.253    45.695  15.078   
 44.724  15.005    43.924  14.982    43.174  14.991    42.534  15.022   
 41.961  15.068    41.403  15.131    40.860  15.208    40.391  15.289   
 39.982  15.371    39.584  15.459    39.198  15.555    38.825  15.658   
 38.463  15.766    38.113  15.879    37.776  15.998    37.451  16.120   
 37.138  16.246    36.837  16.375    36.548  16.507    36.296  16.630   
 36.082  16.738    35.875  16.848    35.676  16.959    35.483  17.071   
 35.298  17.183    35.119  17.295    34.947  17.408    34.782  17.521   
 34.623  17.634    34.470  17.747    34.323  17.860    34.183  17.973   
 34.048  18.085    33.919  18.196    33.795  18.307    33.677  18.418   
 33.564  18.528    33.457  18.637    33.354  18.745    33.256  18.853   
 33.163  18.960    33.074  19.065    32.990  19.170    32.910  19.274   
 32.834  19.377    32.762  19.479    32.695  19.580    32.631  19.680   
 32.571  19.779    32.514  19.877    32.461  19.973    32.411  20.069   
 32.365  20.163    32.321  20.257    32.281  20.349    32.244  20.441   
 32.210  20.531    32.178  20.620    32.149  20.708    32.123  20.795   
 32.099  20.881    32.078  20.966    32.059  21.050    32.043  21.133   
 32.029  21.214    32.016  21.295    32.006  21.375    31.998  21.454   
 31.992  21.532    31.988  21.609    31.986  21.688    31.985  21.782   
 31.988  21.874    31.993  21.977    32.003  22.088    32.019  22.217   
 32.045  22.367    32.090  22.567    32.233  23.001    32.537  23.606   
 32.723  23.891    32.880  24.104    33.024  24.284    33.152  24.433   
 33.275  24.568    33.400  24.701    33.513  24.815    33.616  24.916   
 33.720  25.015    33.825  25.113    33.931  25.208    34.038  25.302   
 34.136  25.385    34.222  25.458    34.308  25.530    34.395  25.600   
 34.482  25.670    34.569  25.739    34.656  25.807    34.744  25.874   
 34.831  25.940    34.918  26.006    35.006  26.071    35.093  26.135   
 35.180  26.198    35.267  26.261    35.354  26.322    35.427  26.374   
 35.496  26.422    35.566  26.471    35.635  26.518    35.704  26.566   
 35.773  26.613    35.841  26.659    35.910  26.706    35.978  26.752   
 36.047  26.797    36.115  26.842    36.183  26.887    36.250  26.932   
 36.318  26.976    36.385  27.020    36.452  27.063    36.519  27.107   
 36.586  27.150    36.652  27.192    36.719  27.235    36.785  27.277   
 36.851  27.318    36.916  27.360    36.981  27.401    37.047  27.442   
 37.111  27.482    37.176  27.523    37.240  27.563    37.304  27.603   
 37.368  27.642    37.432  27.682    37.495  27.721    37.558  27.760   
 37.621  27.798    37.684  27.836    37.746  27.874    37.808  27.912   
 37.870  27.950    37.931  27.987    37.993  28.024    38.054  28.061   
 38.114  28.098    38.175  28.135    38.235  28.171    38.295  28.207   
 38.355  28.243    38.414  28.278    38.473  28.314    38.532  28.349   
 38.590  28.384    38.649  28.419    38.707  28.453    38.765  28.488   
 38.822  28.522    38.879  28.556    38.936  28.590    38.993  28.623   
 39.049  28.657    39.105  28.690    39.161  28.723    39.217  28.756   
 39.272  28.789    39.327  28.821    39.382  28.854    39.437  28.886   
] scurve 
 50.000  16.000 moveto 
[  50.000  16.000    51.405  16.527    52.713  17.131    53.762  17.700   
 54.489  18.143    55.080  18.534    55.577  18.886    56.035  19.229   
 56.387  19.505    56.720  19.778    57.036  20.046    57.334  20.309   
 57.617  20.565    57.846  20.779    58.051  20.975    58.247  21.166   
 58.435  21.352    58.614  21.533    58.786  21.709    58.951  21.880   
 59.110  22.046    59.261  22.207    59.407  22.363    59.547  22.514   
 59.682  22.659    59.812  22.800    59.937  22.936    60.058  23.067   
 60.174  23.194    60.287  23.315    60.396  23.433    60.502  23.546   
 60.605  23.654    60.704  23.758    60.801  23.859    60.895  23.955   
 60.987  24.047    61.077  24.136    61.165  24.221    61.250  24.302   
 61.334  24.380    61.416  24.455    61.496  24.526    61.576  24.595   
 61.653  24.660    61.730  24.722    61.805  24.781    61.879  24.838   
 61.952  24.891    62.024  24.942    62.095  24.991    62.166  25.037   
 62.236  25.081    62.305  25.122    62.373  25.162    62.441  25.199   
 62.508  25.234    62.575  25.266    62.641  25.297    62.707  25.326   
 62.773  25.353    62.838  25.379    62.903  25.402    62.967  25.424   
 63.032  25.444    63.096  25.463    63.160  25.480    63.223  25.495   
 63.287  25.509    63.350  25.521    63.414  25.533    63.483  25.543   
 63.562  25.553    63.650  25.562    63.753  25.570    63.884  25.575   
 64.089  25.574    64.682  25.512    65.021  25.446    65.257  25.388   
 65.458  25.333    65.628  25.282    65.791  25.229    65.944  25.178   
 66.077  25.131    66.211  25.082    66.345  25.031    66.481  24.979   
 66.590  24.935    66.700  24.891    66.810  24.845    66.921  24.798   
 67.033  24.750    67.145  24.701    67.258  24.652    67.372  24.601   
 67.484  24.550    67.576  24.508    67.668  24.465    67.761  24.422   
 67.855  24.378    67.948  24.333    68.043  24.288    68.137  24.242   
 68.232  24.196    68.328  24.149    68.424  24.102    68.521  24.054   
 68.618  24.006    68.715  23.957    68.813  23.908    68.912  23.858   
 69.011  23.808    69.110  23.757    69.210  23.705    69.311  23.654   
 69.412  23.601    69.513  23.548    69.612  23.497    69.694  23.454   
 69.776  23.411    69.859  23.367    69.942  23.324    70.025  23.279   
 70.109  23.235    70.193  23.190    70.277  23.145    70.362  23.100   
 70.447  23.054    70.532  23.008    70.618  22.962    70.704  22.916   
 70.790  22.869    70.877  22.822    70.964  22.775    71.052  22.727   
 71.139  22.679    71.228  22.631    71.316  22.583    71.405  22.534   
 71.495  22.485    71.584  22.436    71.674  22.386    71.765  22.337   
 71.855  22.286    71.947  22.236    72.038  22.186    72.130  22.135   
 72.222  22.083    72.315  22.032    72.408  21.980    72.501  21.928   
 72.595  21.876    72.689  21.824    72.784  21.771    72.879  21.718   
] scurve 
 50.000   7.000 moveto 
[  50.000   7.000    47.992   6.409    45.926   5.959    44.108   5.690   
 42.750   5.565    41.606   5.510    40.574   5.500    39.653   5.523   
 38.858   5.568    38.081   5.634    37.322   5.720    36.636   5.817   
 36.059   5.913    35.496   6.019    34.947   6.135    34.413   6.261   
 33.893   6.395    33.389   6.537    32.901   6.686    32.427   6.841   
 31.969   7.002    31.551   7.160    31.207   7.296    30.873   7.435   
 30.549   7.577    30.234   7.721    29.929   7.867    29.633   8.014   
 29.347   8.163    29.070   8.313    28.801   8.464    28.542   8.617   
 28.291   8.769    28.049   8.923    27.816   9.076    27.590   9.230   
 27.373   9.384    27.163   9.538    26.962   9.691    26.767   9.845   
 26.581   9.997    26.401  10.150    26.229  10.301    26.063  10.452   
 25.904  10.603    25.752  10.752    25.606  10.901    25.467  11.048   
 25.333  11.194    25.206  11.340    25.084  11.484    24.968  11.627   
 24.857  11.769    24.752  11.910    24.652  12.049    24.556  12.187   
 24.466  12.324    24.381  12.460    24.300  12.594    24.223  12.727   
 24.151  12.859    24.084  12.989    24.020  13.118    23.960  13.245   
 23.905  13.371    23.853  13.496    23.804  13.619    23.760  13.741   
 23.718  13.862    23.681  13.981    23.646  14.099    23.614  14.216   
 23.586  14.331    23.561  14.446    23.534  14.581    23.510  14.720   
 23.488  14.879    23.470  15.050    23.457  15.261    23.452  15.534   
 23.486  16.037    23.695  16.977    23.871  17.443    24.026  17.775   
 24.180  18.062    24.312  18.283    24.451  18.498    24.591  18.700   
 24.711  18.863    24.834  19.023    24.961  19.179    25.090  19.333   
 25.221  19.483    25.336  19.609    25.444  19.725    25.554  19.839   
 25.664  19.951    25.776  20.062    25.889  20.171    26.002  20.279   
 26.117  20.385    26.232  20.490    26.347  20.593    26.464  20.695   
 26.580  20.796    26.698  20.895    26.802  20.983    26.897  21.060   
 26.991  21.137    27.086  21.213    27.180  21.289    27.275  21.364   
 27.370  21.438    27.465  21.511    27.560  21.584    27.655  21.656   
 27.750  21.727    27.846  21.798    27.941  21.868    28.036  21.938   
 28.131  22.007    28.226  22.075    28.320  22.143    28.415  22.211   
 28.510  22.277    28.604  22.344    28.699  22.409    28.793  22.475   
 28.887  22.539    28.981  22.604    29.074  22.667    29.168  22.731   
 29.261  22.794    29.354  22.856    29.447  22.918    29.540  22.980   
 29.632  23.041    29.725  23.101    29.817  23.162    29.908  23.221   
 30.000  23.281    30.091  23.340    30.182  23.398    30.273  23.457   
 30.363  23.515    30.453  23.572    30.543  23.629    30.632  23.686   
 30.722  23.742    30.811  23.798    30.899  23.854    30.987  23.909   
 31.076  23.965    31.163  24.019    31.251  24.074    31.338  24.128   
 31.424  24.181    31.511  24.235    31.597  24.288    31.683  24.340   
 31.768  24.393    31.853  24.445    31.938  24.497    32.023  24.548   
 32.107  24.600    32.191  24.651    32.274  24.701    32.357  24.752   
 32.440  24.802    32.523  24.852    32.605  24.901    32.687  24.951   
 32.768  25.000    32.849  25.048    32.930  25.097    33.011  25.145   
 33.091  25.193    33.170  25.241    33.250  25.288    33.329  25.335   
 33.408  25.382    33.486  25.429    33.564  25.475    33.642  25.521   
 33.719  25.567    33.797  25.613    33.873  25.659    33.950  25.704   
 34.026  25.749    34.101  25.793   ] scurve 
[  34.101  25.793    34.177  25.838    34.252  25.882    34.327  25.926   
 34.401  25.970    34.475  26.014    34.549  26.057    34.622  26.100   
 34.695  26.143    34.768  26.186    34.840  26.229    34.912  26.271   
 34.984  26.313    35.056  26.355    35.127  26.396    35.197  26.438   
 35.268  26.479    35.338  26.520    35.408  26.561    35.477  26.602   
 35.546  26.642    35.615  26.682    35.684  26.722   ] scurve 
 50.000   7.000 moveto 
[  50.000   7.000    51.917   7.714    53.714   8.525    55.198   9.307   
 56.256   9.932    57.116  10.484    57.815  10.963    58.475  11.441   
 59.008  11.846    59.482  12.222    59.933  12.593    60.362  12.957   
 60.770  13.315    61.143  13.653    61.440  13.927    61.724  14.196   
 61.997  14.459    62.259  14.717    62.511  14.968    62.753  15.213   
 62.985  15.451    63.208  15.683    63.423  15.909    63.630  16.128   
 63.830  16.341    64.022  16.548    64.207  16.748    64.386  16.942   
 64.559  17.130    64.727  17.311    64.889  17.487    65.046  17.657   
 65.198  17.821    65.345  17.980    65.489  18.133    65.628  18.280   
 65.764  18.422    65.896  18.559    66.025  18.692    66.151  18.819   
 66.274  18.941    66.395  19.059    66.513  19.172    66.628  19.281   
 66.741  19.385    66.852  19.486    66.961  19.582    67.069  19.674   
 67.174  19.762    67.278  19.847    67.381  19.928    67.482  20.006   
 67.582  20.080    67.681  20.151    67.778  20.218    67.875  20.283   
 67.970  20.344    68.065  20.403    68.158  20.458    68.251  20.511   
 68.343  20.561    68.435  20.608    68.526  20.653    68.616  20.695   
 68.706  20.735    68.795  20.773    68.884  20.808    68.973  20.841   
 69.061  20.871    69.148  20.900    69.256  20.932    69.367  20.963   
 69.503  20.996    69.672  21.031    69.923  21.071    70.703  21.110   
 71.176  21.080    71.496  21.040    71.762  20.996    71.986  20.952   
 72.202  20.904    72.403  20.855    72.578  20.809    72.753  20.760   
 72.930  20.708    73.107  20.653    73.250  20.607    73.393  20.560   
 73.537  20.510    73.682  20.459    73.828  20.406    73.974  20.352   
 74.121  20.296    74.268  20.239    74.409  20.183    74.528  20.135   
 74.648  20.086    74.768  20.036    74.889  19.985    75.010  19.934   
 75.132  19.881    75.254  19.828    75.377  19.774    75.501  19.719   
 75.625  19.663    75.749  19.607    75.874  19.550    76.000  19.492   
 76.126  19.433    76.253  19.373    76.380  19.313    76.508  19.252   
 76.637  19.191    76.766  19.128    76.895  19.065    77.014  19.007   
 77.118  18.956    77.223  18.904    77.329  18.852    77.434  18.799   
 77.540  18.746    77.647  18.693    77.754  18.639    77.861  18.584   
 77.969  18.530    78.077  18.474    78.186  18.419    78.294  18.363   
 78.404  18.307    78.514  18.250    78.624  18.193    78.734  18.135   
 78.845  18.077    78.957  18.019    79.068  17.960    79.181  17.901   
 79.293  17.842    79.406  17.782    79.520  17.722    79.634  17.662   
 79.748  17.601    79.863  17.539    79.978  17.478    80.094  17.416   
 80.210  17.354    80.326  17.291    80.443  17.228    80.560  17.164   
 80.678  17.101    80.797  17.037    80.915  16.972    81.034  16.907   
 81.154  16.842    81.274  16.777    81.395  16.711    81.516  16.644   
 81.637  16.578    81.759  16.511    81.881  16.444    82.004  16.376   
 82.127  16.308    82.251  16.240    82.375  16.171    82.500  16.102   
 82.625  16.033    82.751  15.963    82.877  15.893    83.004  15.823   
 83.131  15.752    83.258  15.681    83.386  15.610    83.515  15.538   
 83.644  15.466    83.774  15.393    83.904  15.321    84.034  15.248   
 84.165  15.174    84.297  15.100    84.429  15.026    84.561  14.952   
 84.694  14.877    84.828  14.802    84.962  14.727    85.097  14.651   
 85.232  14.575    85.343  14.512    85.451  14.451    85.560  14.389   
 85.670  14.327    85.780  14.265   ] scurve 
[  85.780  14.265    85.890  14.203    86.000  14.141    86.111  14.078   
 86.222  14.015    86.334  13.952    86.445  13.889    86.558  13.825   
 86.670  13.761    86.783  13.697    86.896  13.633    87.010  13.569   
 87.124  13.504    87.238  13.439    87.352  13.374    87.467  13.309   
 87.583  13.243    87.698  13.177    87.814  13.111    87.931  13.045   
 88.047  12.979    88.164  12.912    88.282  12.845   ] scurve 
 35.000  35.000 moveto 
[  35.000  35.000    35.918  34.138    36.635  33.542    37.114  33.182   
 37.487  32.922    37.827  32.701    38.092  32.539    38.350  32.391   
 38.601  32.254    38.801  32.152    38.992  32.059    39.178  31.974   
 39.360  31.895    39.536  31.823    39.708  31.757    39.853  31.704   
 39.984  31.659    40.111  31.618    40.236  31.580    40.358  31.545   
 40.477  31.513    40.593  31.484    40.706  31.458    40.816  31.434   
 40.924  31.413    41.030  31.394    41.132  31.377    41.233  31.363   
 41.331  31.350    41.426  31.339    41.520  31.331    41.611  31.323   
 41.693  31.318    41.763  31.315    41.832  31.312    41.899  31.311   
 41.965  31.310    42.030  31.310    42.094  31.311    42.157  31.312   
 42.218  31.315    42.279  31.318    42.338  31.321    42.396  31.325   
 42.454  31.330    42.510  31.336    42.566  31.341    42.620  31.348   
 42.674  31.355    42.727  31.362    42.779  31.370    42.830  31.378   
 42.881  31.386    42.930  31.395    42.979  31.405    43.027  31.414   
 43.075  31.424    43.122  31.434    43.168  31.445    43.213  31.455   
 43.258  31.466    43.303  31.478    43.346  31.489    43.389  31.501   
 43.432  31.513    43.474  31.525    43.515  31.537    43.556  31.549   
 43.597  31.562    43.637  31.575    43.676  31.588    43.715  31.601   
 43.754  31.614    43.792  31.627    43.829  31.640    43.867  31.654   
 43.903  31.667    43.940  31.681    43.976  31.694    44.012  31.708   
 44.047  31.722    44.082  31.736    44.117  31.750    44.151  31.763   
 44.185  31.777    44.218  31.791    44.252  31.805    44.285  31.820   
 44.317  31.834    44.350  31.848    44.382  31.862    44.414  31.876   
 44.445  31.890    44.476  31.904    44.507  31.918    44.538  31.933   
 44.569  31.947    44.599  31.961    44.629  31.975    44.659  31.989   
 44.688  32.003    44.717  32.017    44.746  32.031    44.775  32.045   
 44.804  32.059    44.832  32.073    44.860  32.087    44.888  32.101   
 44.916  32.115    44.943  32.129    44.971  32.143    44.998  32.157   
 45.025  32.171    45.052  32.184    45.078  32.198    45.105  32.212   
 45.131  32.225    45.157  32.239    45.183  32.252    45.209  32.266   
 45.234  32.279    45.259  32.293    45.285  32.306    45.310  32.319   
 45.335  32.333    45.359  32.346    45.384  32.359    45.408  32.372   
 45.432  32.385    45.457  32.398    45.481  32.411    45.504  32.424   
 45.528  32.437    45.552  32.450    45.575  32.463    45.598  32.475   
 45.621  32.488    45.644  32.501    45.667  32.513    45.690  32.526   
 45.712  32.538    45.735  32.550    45.757  32.563    45.779  32.575   
 45.801  32.587    45.823  32.600    45.845  32.612    45.866  32.624   
 45.888  32.636    45.909  32.648    45.931  32.660    45.952  32.671   
 45.973  32.683    45.994  32.695    46.015  32.707    46.035  32.718   
 46.056  32.730    46.076  32.741    46.097  32.753    46.117  32.764   
 46.137  32.776    46.157  32.787    46.177  32.798    46.197  32.810   
 46.217  32.821    46.236  32.832    46.256  32.843    46.275  32.854   
 46.294  32.865    46.313  32.876    46.333  32.887    46.352  32.897   
 46.370  32.908    46.389  32.919    46.408  32.929    46.426  32.940   
 46.445  32.951    46.463  32.961    46.481  32.971    46.500  32.982   
 46.518  32.992    46.536  33.002    46.554  33.013    46.571  33.023   
 46.589  33.033    46.607  33.043    46.624  33.053    46.641  33.063   
 46.659  33.073    46.676  33.083   ] scurve 
[  46.676  33.083    46.693  33.093    46.710  33.102    46.727  33.112   
 46.744  33.122    46.761  33.131    46.778  33.141    46.794  33.150   
 46.811  33.160    46.827  33.169    46.843  33.179    46.860  33.188   
 46.876  33.197    46.892  33.207    46.908  33.216    46.924  33.225   
 46.940  33.234    46.955  33.243    46.971  33.252    46.987  33.261   
 47.002  33.270    47.018  33.279    47.033  33.288    47.048  33.296   
 47.063  33.305    47.079  33.314    47.094  33.323    47.109  33.331   
 47.123  33.340    47.138  33.348    47.153  33.357    47.168  33.365   
 47.182  33.374    47.197  33.382    47.211  33.390    47.225  33.399   
 47.240  33.407    47.254  33.415    47.268  33.423    47.282  33.431   
 47.296  33.439    47.310  33.447    47.324  33.455    47.338  33.463   
 47.351  33.471    47.365  33.479    47.378  33.487    47.392  33.494   
 47.405  33.502    47.419  33.510    47.432  33.518    47.445  33.525   
 47.458  33.533    47.471  33.540    47.484  33.548    47.497  33.555   
 47.510  33.563    47.523  33.570    47.536  33.577    47.548  33.585   
 47.561  33.592    47.574  33.599    47.586  33.606    47.598  33.614   
 47.611  33.621    47.623  33.628    47.635  33.635    47.647  33.642   
 47.659  33.649    47.672  33.656    47.683  33.663    47.695  33.670   
 47.707  33.676    47.719  33.683    47.731  33.690    47.742  33.697   
 47.754  33.703    47.766  33.710    47.777  33.717    47.788  33.723   
 47.800  33.730    47.811  33.736    47.822  33.743    47.834  33.749   
 47.845  33.756    47.856  33.762    47.867  33.768    47.878  33.775   
 47.889  33.781    47.900  33.787    47.910  33.794    47.921  33.800   
 47.932  33.806    47.942  33.812    47.953  33.818    47.964  33.824   
 47.974  33.830    47.984  33.836    47.995  33.842    48.005  33.848   
 48.015  33.854    48.026  33.860    48.036  33.866    48.046  33.872   
 48.056  33.878    48.066  33.883   ] scurve 
 35.000  35.000 moveto 
[  35.000  35.000    34.082  35.974    33.363  36.815    32.900  37.390   
 32.545  37.849    32.227  38.270    31.952  38.644    31.681  39.018   
 31.468  39.318    31.256  39.617    31.048  39.916    30.841  40.212   
 30.663  40.470    30.501  40.705    30.340  40.938    30.181  41.170   
 30.023  41.399    29.865  41.627    29.709  41.853    29.554  42.077   
 29.408  42.286    29.285  42.461    29.162  42.635    29.040  42.808   
 28.918  42.979    28.796  43.148    28.675  43.316    28.553  43.482   
 28.432  43.646    28.311  43.809    28.191  43.971    28.070  44.130   
 27.949  44.288    27.828  44.445    27.708  44.600    27.587  44.753   
 27.466  44.905    27.345  45.055    27.223  45.204    27.102  45.351   
 27.002  45.470    26.905  45.586    26.807  45.700    26.709  45.814   
 26.611  45.927    26.513  46.039    26.414  46.151    26.315  46.261   
 26.216  46.370    26.117  46.479    26.017  46.587    25.918  46.694   
 25.817  46.800    25.717  46.906    25.616  47.011    25.515  47.115   
 25.414  47.218    25.312  47.321    25.211  47.423    25.108  47.524   
 25.006  47.625    24.903  47.725    24.799  47.824    24.695  47.923   
 24.591  48.022    24.487  48.119    24.382  48.216    24.277  48.313   
 24.171  48.409    24.065  48.505    23.959  48.600    23.852  48.694   
 23.744  48.788    23.637  48.882    23.529  48.975    23.420  49.068   
 23.311  49.160    23.202  49.252    23.092  49.344    22.981  49.435   
 22.871  49.526    22.760  49.617    22.648  49.707    22.536  49.797   
 22.423  49.887    22.310  49.976    22.197  50.066    22.083  50.154   
 21.968  50.243    21.853  50.332    21.738  50.420    21.622  50.508   
 21.506  50.595    21.389  50.683    21.272  50.771    21.154  50.858   
 21.036  50.945    20.917  51.032    20.797  51.119    20.678  51.205   
 20.557  51.292    20.437  51.379    20.315  51.465    20.193  51.551   
 20.091  51.623    19.993  51.692    19.894  51.761    19.796  51.830   
 19.696  51.899    19.597  51.968    19.497  52.037    19.397  52.106   
 19.296  52.174    19.195  52.243    19.094  52.312    18.993  52.381   
 18.891  52.450    18.789  52.518    18.686  52.587    18.583  52.656   
 18.480  52.725    18.376  52.794    18.272  52.863    18.168  52.932   
 18.063  53.001    17.958  53.070    17.853  53.139    17.747  53.208   
 17.641  53.277    17.535  53.347    17.428  53.416    17.321  53.485   
 17.213  53.555    17.105  53.624    16.997  53.694    16.889  53.764   
 16.780  53.833    16.670  53.903    16.561  53.973    16.450  54.043   
 16.340  54.113    16.229  54.183    16.118  54.254    16.006  54.324   
 15.895  54.395    15.782  54.465    15.670  54.536    15.557  54.607   
 15.443  54.678    15.329  54.749    15.215  54.820    15.100  54.891   
 14.985  54.963    14.870  55.034    14.754  55.106    14.638  55.178   
 14.522  55.250    14.405  55.322    14.288  55.394    14.170  55.466   
 14.052  55.539    13.933  55.611    13.814  55.684    13.695  55.757   
 13.576  55.830    13.455  55.903    13.335  55.977    13.214  56.050   
 13.093  56.124    12.971  56.198    12.849  56.272    12.727  56.346   
 12.604  56.421    12.480  56.495    12.357  56.570    12.233  56.645   
 12.108  56.720    11.983  56.795    11.858  56.871    11.732  56.946   
 11.606  57.022    11.479  57.098    11.352  57.174    11.224  57.251   
 11.097  57.327    10.968  57.404    10.839  57.481    10.710  57.558   
 10.581  57.635    10.451  57.713   ] scurve 
[  10.451  57.713    10.320  57.791    10.189  57.869    10.058  57.947   
  9.926  58.025     9.794  58.104     9.661  58.183     9.528  58.262   
  9.395  58.341     9.261  58.420     9.126  58.500     8.991  58.580   
  8.856  58.660     8.720  58.740     8.584  58.821     8.447  58.902   
  8.310  58.983     8.172  59.064     8.034  59.145     7.896  59.227   
  7.757  59.309     7.618  59.391     7.478  59.473     7.337  59.556   
  7.196  59.639     7.055  59.722     6.913  59.805     6.771  59.889   
  6.628  59.973     6.485  60.057     6.342  60.141     6.198  60.226   
  6.053  60.311     5.908  60.396     5.762  60.481     5.616  60.567   
  5.470  60.652     5.323  60.738     5.175  60.825     5.027  60.911   
  4.879  60.998     4.730  61.085   ] scurve 
 20.000  35.000 moveto 
[  20.000  35.000    21.379  33.684    22.716  32.528    23.666  31.779   
 24.410  31.234    25.062  30.785    25.583  30.447    26.082  30.139   
 26.569  29.855    26.949  29.644    27.322  29.446    27.687  29.262   
 28.043  29.091    28.391  28.932    28.705  28.796    28.971  28.686   
 29.232  28.583    29.487  28.487    29.736  28.397    29.981  28.314   
 30.220  28.236    30.453  28.165    30.682  28.099    30.906  28.038   
 31.125  27.983    31.338  27.932    31.548  27.886    31.752  27.844   
 31.945  27.808    32.101  27.780    32.255  27.756    32.406  27.733   
 32.555  27.713    32.700  27.695    32.844  27.680    32.984  27.666   
 33.122  27.654    33.258  27.644    33.392  27.636    33.523  27.630   
 33.651  27.625    33.778  27.622    33.902  27.620    34.024  27.620   
 34.145  27.621    34.263  27.623    34.379  27.627    34.493  27.632   
 34.606  27.638    34.717  27.645    34.826  27.653    34.933  27.662   
 35.038  27.673    35.142  27.684    35.244  27.696    35.345  27.709   
 35.444  27.722    35.542  27.737    35.638  27.752    35.733  27.768   
 35.827  27.784    35.919  27.801    36.010  27.819    36.100  27.837   
 36.188  27.856    36.276  27.875    36.362  27.895    36.447  27.916   
 36.531  27.936    36.613  27.957    36.695  27.979    36.776  28.001   
 36.856  28.023    36.935  28.046    37.013  28.069    37.090  28.092   
 37.166  28.115    37.241  28.139    37.315  28.163    37.389  28.187   
 37.462  28.212    37.534  28.236    37.605  28.261    37.675  28.286   
 37.745  28.311    37.814  28.337    37.882  28.362    37.950  28.388   
 38.017  28.413    38.083  28.439    38.149  28.465    38.214  28.491   
 38.278  28.517    38.342  28.543    38.406  28.570    38.468  28.596   
 38.530  28.622    38.592  28.649    38.653  28.675    38.714  28.702   
 38.774  28.728    38.833  28.755    38.892  28.781    38.951  28.808   
 39.009  28.834    39.067  28.861    39.124  28.887    39.181  28.914   
 39.237  28.940    39.293  28.967    39.349  28.993    39.404  29.020   
 39.458  29.046    39.513  29.072    39.566  29.099    39.620  29.125   
 39.673  29.151    39.726  29.177    39.778  29.203    39.830  29.229   
 39.882  29.255    39.933  29.281    39.984  29.307    40.035  29.333   
 40.085  29.359    40.135  29.385    40.185  29.410    40.234  29.436   
 40.283  29.461    40.332  29.487    40.380  29.512    40.428  29.537   
 40.476  29.563    40.524  29.588    40.571  29.613    40.618  29.638   
 40.664  29.663    40.711  29.687    40.757  29.712    40.803  29.737   
 40.848  29.761    40.894  29.786    40.939  29.810    40.983  29.834   
 41.028  29.859    41.072  29.883    41.116  29.907    41.160  29.931   
 41.204  29.954    41.247  29.978    41.290  30.002    41.333  30.025   
 41.375  30.049    41.418  30.072    41.460  30.095    41.501  30.119   
 41.543  30.142    41.584  30.165    41.626  30.188    41.667  30.210   
 41.707  30.233    41.748  30.256    41.788  30.278    41.828  30.301   
 41.868  30.323    41.908  30.345    41.947  30.367    41.987  30.389   
 42.026  30.411    42.064  30.433    42.103  30.455    42.141  30.476   
 42.180  30.498    42.218  30.519    42.256  30.541    42.293  30.562   
 42.331  30.583    42.368  30.604    42.405  30.625    42.442  30.646   
 42.479  30.667    42.515  30.688    42.551  30.708    42.588  30.729   
 42.623  30.749    42.659  30.770    42.695  30.790    42.730  30.810   
 42.765  30.830    42.801  30.850   ] scurve 
[  42.801  30.850    42.835  30.870    42.870  30.890    42.905  30.909   
 42.939  30.929    42.973  30.949    43.007  30.968    43.041  30.987   
 43.075  31.007    43.108  31.026    43.142  31.045    43.175  31.064   
 43.208  31.083    43.241  31.101    43.273  31.120    43.306  31.139   
 43.338  31.157    43.370  31.176    43.402  31.194    43.434  31.212   
 43.466  31.231    43.498  31.249    43.529  31.267    43.560  31.285   
 43.591  31.303    43.622  31.320    43.653  31.338    43.684  31.356   
 43.714  31.373    43.745  31.391    43.775  31.408    43.805  31.425   
 43.835  31.442    43.865  31.460   ] scurve 
 20.000  35.000 moveto 
[  20.000  35.000    18.621  36.443    17.306  37.935    16.370  39.065   
 15.680  39.934    15.087  40.701    14.562  41.397    14.092  42.033   
 13.682  42.594    13.278  43.156    12.877  43.718    12.553  44.177   
 12.240  44.625    11.928  45.071    11.620  45.516    11.314  45.958   
 11.010  46.399    10.747  46.781    10.507  47.129    10.269  47.475   
 10.032  47.819     9.796  48.161     9.561  48.501     9.327  48.838   
  9.094  49.173     8.862  49.505     8.631  49.835     8.401  50.162   
  8.172  50.487     7.980  50.756     7.797  51.012     7.615  51.266   
  7.433  51.518     7.251  51.768     7.069  52.017     6.887  52.263   
  6.706  52.508     6.525  52.751     6.344  52.993     6.162  53.232   
  5.981  53.470     5.800  53.705     5.619  53.939     5.438  54.172   
  5.257  54.402     5.075  54.631     4.894  54.858     4.712  55.083   
] scurve 
  5.000  35.000 moveto 
[   5.000  35.000     6.656  33.404     8.299  31.942     9.769  30.733   
 10.876  29.886    11.817  29.208    12.598  28.674    13.363  28.177   
 13.976  27.797    14.565  27.448    15.142  27.121    15.707  26.814   
 16.200  26.558    16.635  26.342    17.061  26.138    17.479  25.946   
 17.890  25.765    18.292  25.596    18.686  25.437    19.072  25.288   
 19.450  25.150    19.813  25.023    20.103  24.927    20.388  24.836   
 20.668  24.751    20.944  24.670    21.214  24.596    21.480  24.526   
 21.741  24.460    21.997  24.400    22.248  24.344    22.496  24.292   
 22.738  24.244    22.977  24.201    23.211  24.161    23.440  24.125   
 23.666  24.092    23.888  24.063    24.105  24.037    24.319  24.014   
 24.529  23.994    24.736  23.977    24.938  23.963    25.137  23.951   
 25.333  23.942    25.525  23.935    25.714  23.931    25.900  23.929   
 26.083  23.929    26.262  23.931    26.439  23.936    26.613  23.942   
 26.783  23.949    26.951  23.959    27.117  23.970    27.279  23.983   
 27.440  23.997    27.597  24.013    27.752  24.030    27.905  24.048   
 28.056  24.068    28.204  24.088    28.350  24.110    28.494  24.133   
 28.636  24.157    28.776  24.182    28.914  24.208    29.050  24.235   
 29.184  24.263    29.316  24.291    29.446  24.320    29.575  24.350   
 29.698  24.380    29.798  24.405    29.898  24.430    29.996  24.456   
 30.094  24.482    30.190  24.508    30.286  24.535    30.381  24.562   
 30.475  24.589    30.567  24.617    30.659  24.645    30.751  24.673   
 30.841  24.702    30.931  24.730    31.019  24.759    31.107  24.788   
 31.194  24.818    31.281  24.847    31.367  24.877    31.452  24.907   
 31.536  24.937    31.619  24.967    31.702  24.998    31.784  25.028   
 31.866  25.059    31.947  25.089    32.027  25.120    32.107  25.151   
 32.186  25.182    32.264  25.214    32.342  25.245    32.419  25.276   
 32.496  25.308    32.572  25.339    32.648  25.371    32.723  25.402   
 32.797  25.434    32.871  25.465    32.945  25.497    33.018  25.529   
 33.090  25.561    33.162  25.592    33.234  25.624    33.305  25.656   
 33.375  25.688    33.446  25.720    33.515  25.752    33.584  25.783   
 33.653  25.815    33.722  25.847    33.790  25.879    33.857  25.911   
 33.924  25.943    33.991  25.974    34.057  26.006    34.123  26.038   
 34.189  26.069    34.254  26.101    34.319  26.133    34.383  26.164   
 34.447  26.196    34.511  26.227    34.574  26.259    34.637  26.290   
 34.700  26.321    34.762  26.353    34.824  26.384    34.886  26.415   
 34.947  26.446    35.008  26.477    35.069  26.508    35.129  26.539   
 35.189  26.570    35.249  26.601    35.308  26.631    35.367  26.662   
 35.426  26.693    35.484  26.723    35.543  26.754    35.600  26.784   
 35.658  26.814    35.715  26.845    35.772  26.875    35.829  26.905   
 35.886  26.935    35.942  26.965    35.998  26.994    36.054  27.024   
 36.109  27.054    36.164  27.084    36.219  27.113    36.274  27.142   
 36.328  27.172    36.382  27.201    36.436  27.230    36.490  27.259   
 36.543  27.288    36.596  27.317    36.649  27.346    36.702  27.375   
 36.754  27.404    36.806  27.432    36.858  27.461    36.910  27.489   
 36.962  27.517    37.013  27.546    37.064  27.574    37.115  27.602   
 37.165  27.630    37.216  27.658    37.266  27.685    37.316  27.713   
 37.365  27.741    37.415  27.768    37.464  27.795    37.513  27.823   
 37.562  27.850    37.611  27.877   ] scurve 
[  37.611  27.877    37.659  27.904    37.707  27.931    37.755  27.958   
 37.803  27.985    37.851  28.011    37.898  28.038    37.945  28.065   
 37.992  28.091    38.039  28.117    38.086  28.143    38.132  28.170   
 38.179  28.196    38.225  28.222    38.270  28.247    38.316  28.273   
 38.362  28.299    38.407  28.324    38.452  28.350    38.497  28.375   
 38.542  28.401    38.586  28.426    38.631  28.451    38.675  28.476   
 38.719  28.501    38.763  28.526    38.806  28.551    38.850  28.575   
 38.893  28.600    38.936  28.624    38.979  28.649    39.022  28.673   
 39.064  28.697    39.107  28.722    39.149  28.746    39.191  28.770   
 39.233  28.793    39.275  28.817    39.316  28.841    39.358  28.865   
 39.399  28.888    39.440  28.912    39.481  28.935    39.522  28.958   
 39.562  28.981    39.603  29.005    39.643  29.028    39.683  29.050   
 39.723  29.073    39.763  29.096    39.803  29.119    39.842  29.141   
 39.881  29.164    39.920  29.186    39.959  29.209    39.998  29.231   
 40.037  29.253    40.076  29.275    40.114  29.297    40.152  29.319   
 40.190  29.341    40.228  29.363    40.266  29.384    40.304  29.406   
 40.341  29.427    40.378  29.449    40.416  29.470    40.453  29.491   
 40.490  29.513    40.526  29.534    40.563  29.555    40.599  29.576   
] scurve 
  5.000  35.000 moveto 
[   5.000  35.000     3.344  36.718     1.700  38.542   ] scurve 
 50.000  45.000 moveto 
[  50.000  45.000    50.849  45.244    51.723  45.417    52.443  45.502   
 52.997  45.532    53.432  45.535    53.856  45.519    54.201  45.493   
 54.521  45.458    54.832  45.413    55.132  45.360    55.401  45.303   
 55.625  45.250    55.841  45.192    56.050  45.130    56.251  45.065   
 56.444  44.998    56.630  44.927    56.808  44.854    56.979  44.780   
 57.142  44.703    57.298  44.626    57.446  44.547    57.588  44.467   
 57.723  44.387    57.851  44.306    57.972  44.225    58.087  44.144   
 58.196  44.063    58.300  43.982    58.397  43.902    58.489  43.822   
 58.575  43.743    58.656  43.664    58.733  43.586    58.804  43.508   
 58.871  43.432    58.933  43.356    58.992  43.281    59.046  43.208   
 59.096  43.135    59.142  43.063    59.185  42.992    59.224  42.923   
 59.260  42.854    59.293  42.786    59.322  42.720    59.349  42.654   
 59.373  42.590    59.394  42.527    59.413  42.464    59.429  42.403   
 59.443  42.343    59.454  42.283    59.464  42.225    59.471  42.168   
 59.476  42.111    59.480  42.056    59.482  42.002    59.482  41.948   
 59.480  41.895    59.477  41.844    59.472  41.793    59.466  41.743   
 59.458  41.693    59.449  41.645    59.439  41.597    59.428  41.550   
 59.416  41.504    59.402  41.459    59.388  41.414    59.372  41.370   
 59.356  41.327    59.338  41.284    59.315  41.232    59.291  41.180   
 59.261  41.121    59.226  41.056    59.180  40.978    59.115  40.877   
 58.954  40.663    58.693  40.375    58.544  40.232    58.425  40.125   
 58.329  40.042    58.232  39.962    58.154  39.900    58.076  39.838   
 57.997  39.778    57.928  39.726    57.865  39.680    57.803  39.634   
 57.740  39.589    57.678  39.545    57.616  39.502    57.554  39.459   
 57.503  39.424    57.454  39.391    57.404  39.358    57.355  39.325   
 57.307  39.292    57.258  39.260    57.209  39.228    57.161  39.197   
 57.113  39.166    57.065  39.135    57.017  39.104    56.970  39.074   
 56.923  39.044    56.876  39.014    56.829  38.985    56.784  38.957   
 56.747  38.934    56.710  38.911    56.674  38.888    56.637  38.865   
 56.600  38.843    56.564  38.820    56.528  38.798    56.492  38.776   
 56.456  38.754    56.421  38.733   ] scurve 
 50.000  45.000 moveto 
[  50.000  45.000    49.197  44.695    48.455  44.344    47.898  44.033   
 47.500  43.783    47.192  43.573    46.916  43.371    46.682  43.190   
 46.489  43.033    46.307  42.879    46.136  42.728    45.974  42.582   
 45.828  42.444    45.712  42.333    45.602  42.225    45.498  42.120   
 45.398  42.018    45.302  41.919    45.211  41.824    45.124  41.731   
 45.041  41.642    44.961  41.556    44.884  41.473    44.810  41.393   
 44.740  41.317    44.672  41.243    44.606  41.172    44.543  41.104   
 44.482  41.038    44.423  40.976    44.366  40.916    44.310  40.859   
 44.256  40.804    44.204  40.751    44.153  40.701    44.103  40.653   
 44.054  40.607    44.007  40.564    43.961  40.522    43.915  40.483   
 43.870  40.445    43.826  40.409    43.783  40.376    43.741  40.343   
 43.699  40.313    43.658  40.284    43.617  40.257    43.577  40.231   
 43.537  40.207    43.498  40.184    43.459  40.162    43.420  40.142   
 43.382  40.123    43.344  40.105    43.306  40.089    43.269  40.074   
 43.231  40.059    43.194  40.046    43.157  40.034    43.120  40.023   
 43.084  40.013    43.047  40.004    43.011  39.996    42.974  39.989   
 42.938  39.982    42.901  39.977    42.865  39.972    42.829  39.968   
 42.793  39.965    42.757  39.963    42.720  39.961    42.684  39.960   
 42.648  39.960    42.612  39.960    42.576  39.961    42.539  39.963   
 42.503  39.965    42.467  39.968    42.422  39.973    42.376  39.978   
 42.322  39.986    42.259  39.995    42.179  40.010    42.050  40.039   
 41.692  40.141    41.483  40.214    41.338  40.270    41.215  40.319   
 41.109  40.363    41.009  40.406    40.913  40.448    40.832  40.484   
 40.750  40.522    40.667  40.561    40.583  40.600    40.511  40.634   
 40.443  40.667    40.375  40.700    40.306  40.734    40.237  40.768   
 40.167  40.803    40.097  40.839    40.026  40.874    39.955  40.911   
 39.891  40.943    39.833  40.973    39.775  41.003    39.717  41.034   
 39.658  41.064    39.599  41.095    39.540  41.126    39.480  41.158   
 39.420  41.190    39.360  41.222    39.300  41.254    39.239  41.287   
 39.177  41.320    39.116  41.353    39.054  41.387    38.992  41.420   
 38.929  41.454    38.866  41.489    38.803  41.523    38.739  41.558   
 38.675  41.593    38.610  41.629    38.546  41.664    38.480  41.700   
 38.415  41.736    38.351  41.772    38.298  41.801    38.245  41.831   
 38.192  41.860    38.138  41.890    38.084  41.920    38.030  41.950   
 37.976  41.981    37.921  42.011    37.866  42.042    37.811  42.073   
 37.756  42.104    37.700  42.135    37.644  42.167    37.588  42.198   
 37.532  42.230    37.475  42.262    37.418  42.294    37.361  42.326   
 37.303  42.358    37.246  42.391    37.188  42.424    37.130  42.457   
 37.071  42.490    37.012  42.523    36.953  42.556    36.894  42.590   
 36.835  42.624    36.775  42.658    36.715  42.692    36.654  42.726   
 36.594  42.761    36.533  42.795    36.472  42.830    36.410  42.865   
 36.349  42.900    36.287  42.935    36.224  42.971    36.162  43.007   
 36.099  43.042    36.036  43.078    35.972  43.115    35.909  43.151   
 35.845  43.188    35.780  43.224    35.716  43.261    35.651  43.298   
 35.586  43.336    35.520  43.373    35.455  43.411    35.389  43.449   
 35.322  43.487    35.256  43.525    35.189  43.563    35.121  43.602   
 35.054  43.640    34.986  43.679    34.918  43.718    34.849  43.758   
 34.781  43.797    34.711  43.837   ] scurve 
[  34.711  43.837    34.642  43.877    34.572  43.917    34.502  43.957   
 34.432  43.997    34.361  44.038    34.290  44.079    34.219  44.120   
 34.147  44.161    34.075  44.202    34.003  44.244    33.930  44.285   
 33.857  44.327    33.784  44.369    33.711  44.412    33.637  44.454   
 33.562  44.497    33.488  44.540    33.413  44.583    33.337  44.626   
 33.262  44.670    33.186  44.714    33.109  44.758    33.033  44.802   
 32.956  44.846    32.878  44.891    32.801  44.935    32.723  44.980   
 32.644  45.026    32.565  45.071    32.486  45.117    32.407  45.162   
 32.327  45.208    32.247  45.255    32.166  45.301    32.085  45.348   
 32.004  45.395    31.922  45.442    31.840  45.489    31.758  45.536   
 31.675  45.584    31.592  45.632    31.508  45.680    31.424  45.729   
 31.340  45.777    31.255  45.826    31.170  45.875    31.085  45.925   
 30.999  45.974    30.912  46.024    30.826  46.074    30.739  46.124   
 30.651  46.174    30.563  46.225    30.475  46.276    30.387  46.327   
 30.298  46.378    30.208  46.430    30.118  46.482    30.028  46.534   
 29.937  46.586    29.846  46.639    29.755  46.692    29.663  46.745   
 29.571  46.798    29.478  46.851    29.385  46.905    29.291  46.959   
 29.197  47.013    29.111  47.063    29.036  47.107    28.959  47.150   
 28.883  47.195    28.806  47.239    28.730  47.283    28.652  47.328   
 28.575  47.372    28.497  47.417    28.419  47.462    28.341  47.508   
 28.262  47.553    28.183  47.598    28.104  47.644    28.025  47.690   
 27.945  47.736    27.865  47.782    27.784  47.829    27.704  47.875   
 27.623  47.922    27.542  47.969    27.460  48.016    27.378  48.063   
 27.296  48.110    27.214  48.158    27.131  48.206    27.048  48.254   
 26.965  48.302    26.881  48.350    26.797  48.398    26.713  48.447   
 26.628  48.496    26.544  48.545    26.459  48.594    26.373  48.643   
 26.287  48.693    26.201  48.742    26.115  48.792    26.028  48.842   
 25.941  48.893    25.854  48.943    25.766  48.994    25.678  49.044   
 25.590  49.095    25.501  49.146    25.412  49.198    25.323  49.249   
 25.234  49.301    25.144  49.353    25.054  49.405    24.963  49.457   
 24.872  49.510    24.781  49.562    24.689  49.615    24.597  49.668   
 24.505  49.721    24.413  49.775    24.320  49.829    24.227  49.882   
 24.133  49.936    24.039  49.991    23.945  50.045    23.850  50.100   
 23.756  50.154    23.660  50.209    23.565  50.265    23.469  50.320   
 23.372  50.376    23.276  50.431    23.179  50.487    23.081  50.544   
 22.984  50.600    22.886  50.657    22.787  50.713    22.689  50.770   
 22.589  50.828    22.490  50.885    22.390  50.943    22.290  51.001   
 22.189  51.059    22.088  51.117    21.987  51.176    21.885  51.234   
 21.783  51.293    21.681  51.352    21.578  51.412    21.475  51.471   
 21.371  51.531    21.267  51.591    21.163  51.651    21.059  51.712   
 20.953  51.772    20.848  51.833    20.742  51.894    20.636  51.956   
 20.529  52.017    20.423  52.079    20.315  52.141    20.207  52.203   
 20.099  52.266    19.991  52.328    19.882  52.391    19.773  52.454   
 19.663  52.518    19.553  52.581    19.442  52.645    19.331  52.709   
 19.220  52.773    19.108  52.838    18.996  52.903    18.884  52.968   
 18.771  53.033    18.657  53.098    18.544  53.164    18.429  53.230   
 18.315  53.296    18.200  53.362    18.084  53.429    17.969  53.496   
 17.852  53.563    17.736  53.631   ] scurve 
[  17.736  53.631    17.618  53.698    17.501  53.766    17.383  53.834   
 17.265  53.903    17.146  53.971    17.027  54.040    16.907  54.109   
 16.787  54.178    16.666  54.248    16.545  54.318    16.424  54.388   
 16.302  54.458    16.180  54.529    16.057  54.600    15.934  54.671   
 15.810  54.743    15.686  54.814    15.561  54.886    15.436  54.958   
 15.311  55.031    15.185  55.103    15.059  55.176    14.932  55.250   
 14.805  55.323    14.677  55.397    14.549  55.471    14.420  55.545   
 14.291  55.620    14.161  55.695    14.031  55.770    13.901  55.845   
 13.770  55.921    13.638  55.997    13.506  56.073    13.374  56.150   
 13.241  56.226    13.107  56.303    12.973  56.381    12.839  56.458   
 12.704  56.536    12.569  56.614    12.433  56.693    12.297  56.772   
 12.160  56.851    12.022  56.930    11.884  57.010    11.746  57.090   
 11.607  57.170    11.468  57.250    11.328  57.331    11.188  57.412   
 11.047  57.493    10.921  57.566    10.808  57.632    10.694  57.697   
 10.580  57.763    10.465  57.829    10.350  57.895    10.235  57.962   
 10.120  58.029    10.004  58.095     9.888  58.162     9.772  58.230   
  9.655  58.297     9.538  58.365     9.420  58.433     9.303  58.501   
  9.184  58.569     9.066  58.637     8.947  58.706     8.828  58.775   
  8.709  58.844     8.589  58.913     8.468  58.982     8.348  59.052   
  8.227  59.122     8.106  59.192     7.984  59.262     7.862  59.332   
  7.740  59.403     7.617  59.474     7.494  59.545     7.371  59.616   
  7.247  59.687     7.123  59.759     6.999  59.831     6.874  59.903   
  6.749  59.975     6.623  60.048     6.497  60.121     6.371  60.193   
  6.244  60.267     6.117  60.340     5.990  60.413     5.862  60.487   
  5.734  60.561     5.605  60.635     5.477  60.710     5.347  60.784   
  5.218  60.859     5.088  60.934     4.957  61.010     4.827  61.085   
] scurve 
 50.000  54.000 moveto 
[  50.000  54.000    51.477  54.430    52.998  54.747    54.305  54.922   
 55.276  54.995    56.076  55.018    56.826  55.009    57.466  54.978   
 58.039  54.932    58.597  54.869    59.140  54.792    59.609  54.711   
 60.018  54.629    60.416  54.541    60.802  54.445    61.175  54.342   
 61.537  54.234    61.887  54.121    62.224  54.002    62.549  53.880   
 62.862  53.754    63.163  53.625    63.452  53.493    63.704  53.370   
 63.918  53.262    64.125  53.152    64.324  53.041    64.517  52.929   
 64.702  52.817    64.881  52.705    65.053  52.592    65.218  52.479   
 65.377  52.366    65.530  52.253    65.677  52.140    65.817  52.027   
 65.952  51.915    66.081  51.804    66.205  51.693    66.323  51.582   
 66.436  51.472    66.543  51.363    66.646  51.255    66.744  51.147   
 66.837  51.040    66.926  50.935    67.010  50.830    67.090  50.726   
 67.166  50.623    67.238  50.521    67.305  50.420    67.369  50.320   
 67.429  50.221    67.486  50.123    67.539  50.027    67.589  49.931   
 67.635  49.837    67.679  49.743    67.719  49.651    67.756  49.559   
 67.790  49.469    67.822  49.380    67.851  49.292    67.877  49.205   
 67.901  49.119    67.922  49.034    67.941  48.950    67.957  48.867   
 67.971  48.786    67.984  48.705    67.994  48.625    68.002  48.546   
 68.008  48.468    68.012  48.391    68.014  48.312    68.015  48.218   
 68.012  48.126    68.007  48.023    67.997  47.912    67.981  47.783   
 67.955  47.633    67.910  47.433    67.767  46.999    67.463  46.394   
 67.277  46.109    67.120  45.896    66.976  45.716    66.848  45.567   
 66.725  45.432    66.600  45.299    66.487  45.185    66.384  45.084   
 66.280  44.985    66.175  44.887    66.069  44.792    65.962  44.698   
 65.864  44.615    65.778  44.542    65.692  44.470    65.605  44.400   
 65.518  44.330    65.431  44.261    65.344  44.193    65.256  44.126   
 65.169  44.060    65.082  43.994    64.994  43.929    64.907  43.865   
 64.820  43.802    64.733  43.739    64.646  43.678    64.573  43.626   
 64.504  43.578    64.434  43.529    64.365  43.482    64.296  43.434   
 64.227  43.387    64.159  43.341    64.090  43.294    64.022  43.248   
 63.953  43.203    63.885  43.158    63.817  43.113    63.750  43.068   
 63.682  43.024    63.615  42.980    63.548  42.937    63.481  42.893   
 63.414  42.850    63.348  42.808    63.281  42.765    63.215  42.723   
 63.149  42.682    63.084  42.640    63.019  42.599    62.953  42.558   
 62.889  42.518    62.824  42.477    62.760  42.437    62.696  42.397   
 62.632  42.358    62.568  42.318    62.505  42.279    62.442  42.240   
 62.379  42.202    62.316  42.164    62.254  42.126    62.192  42.088   
 62.130  42.050    62.069  42.013    62.007  41.976    61.946  41.939   
 61.886  41.902    61.825  41.865    61.765  41.829    61.705  41.793   
 61.645  41.757    61.586  41.722    61.527  41.686    61.468  41.651   
 61.410  41.616    61.351  41.581    61.293  41.547    61.235  41.512   
 61.178  41.478    61.121  41.444    61.064  41.410    61.007  41.377   
 60.951  41.343    60.895  41.310    60.839  41.277    60.783  41.244   
 60.728  41.211    60.673  41.179    60.618  41.146    60.563  41.114   
 60.509  41.082    60.455  41.051    60.401  41.019    60.348  40.988   
 60.294  40.956    60.241  40.925    60.189  40.894    60.136  40.864   
 60.084  40.833    60.032  40.803    59.981  40.772    59.929  40.742   
 59.878  40.712    59.827  40.683   ] scurve 
[  59.827  40.683    59.776  40.653    59.726  40.624    59.676  40.594   
 59.626  40.565    59.576  40.536    59.527  40.507    59.478  40.479   
 59.429  40.450    59.380  40.422    59.332  40.394    59.283  40.366   
 59.236  40.338    59.188  40.310    59.140  40.282    59.093  40.255   
 59.046  40.228    58.999  40.200    58.953  40.173    58.907  40.147   
 58.861  40.120    58.815  40.093    58.769  40.067    58.724  40.040   
 58.679  40.014    58.634  39.988    58.589  39.962    58.545  39.937   
 58.501  39.911    58.457  39.886    58.413  39.860    58.370  39.835   
 58.326  39.810    58.283  39.785    58.240  39.760    58.198  39.735   
 58.155  39.711    58.113  39.686    58.071  39.662    58.030  39.638   
 57.988  39.614    57.947  39.590    57.906  39.566    57.865  39.542   
 57.824  39.519    57.783  39.495    57.743  39.472    57.703  39.449   
 57.663  39.426    57.624  39.403    57.584  39.380    57.545  39.357   
 57.506  39.335    57.467  39.312    57.428  39.290    57.390  39.268   
 57.352  39.246    57.314  39.224    57.276  39.202    57.238  39.180   
 57.201  39.158    57.163  39.137    57.126  39.115    57.089  39.094   
 57.053  39.073    57.016  39.052    56.980  39.031    56.944  39.010   
 56.908  38.989    56.872  38.968    56.836  38.948    56.801  38.927   
 56.766  38.907    56.731  38.887    56.696  38.866    56.661  38.846   
 56.627  38.826    56.592  38.807    56.558  38.787    56.524  38.767   
 56.490  38.748    56.457  38.728    56.423  38.709    56.390  38.690   
 56.357  38.671    56.324  38.652    56.291  38.633    56.259  38.614   
 56.226  38.595    56.194  38.577    56.162  38.558    56.130  38.540   
 56.098  38.521    56.067  38.503    56.035  38.485    56.004  38.467   
 55.973  38.449    55.942  38.431    55.911  38.413    55.881  38.396   
 55.850  38.378    55.820  38.360    55.790  38.343    55.760  38.326   
 55.730  38.308    55.700  38.291    55.671  38.274    55.641  38.257   
 55.612  38.240    55.583  38.224    55.554  38.207    55.525  38.190   
 55.497  38.174    55.468  38.157    55.440  38.141    55.412  38.125   
 55.384  38.109    55.356  38.092    55.328  38.076    55.301  38.060   
 55.273  38.045    55.246  38.029    55.219  38.013    55.192  37.998   
 55.165  37.982    55.138  37.967    55.111  37.951    55.085  37.936   
 55.059  37.921    55.032  37.906    55.006  37.891    54.980  37.876   
 54.955  37.861    54.929  37.846    54.903  37.831    54.878  37.816   
 54.853  37.802    54.828  37.787    54.803  37.773    54.778  37.759   
 54.753  37.744    54.728  37.730    54.704  37.716    54.679  37.702   
 54.655  37.688    54.631  37.674    54.607  37.660    54.583  37.646   
 54.560  37.633    54.536  37.619    54.512  37.605    54.489  37.592   
 54.466  37.578    54.443  37.565    54.420  37.552    54.397  37.539   
 54.374  37.525    54.351  37.512    54.329  37.499    54.306  37.486   
 54.284  37.473    54.262  37.461    54.240  37.448    54.218  37.435   
 54.196  37.423    54.174  37.410    54.153  37.398    54.131  37.385   
 54.110  37.373    54.088  37.361    54.067  37.348    54.046  37.336   
 54.025  37.324    54.004  37.312    53.984  37.300    53.963  37.288   
 53.942  37.276    53.922  37.264    53.902  37.253    53.881  37.241   
 53.861  37.229    53.841  37.218    53.821  37.206    53.802  37.195   
 53.782  37.184    53.762  37.172    53.743  37.161    53.724  37.150   
 53.704  37.139    53.685  37.128   ] scurve 
[  53.685  37.128    53.666  37.117    53.647  37.106    53.628  37.095   
 53.609  37.084    53.591  37.073    53.572  37.062    53.553  37.052   
 53.535  37.041    53.517  37.030    53.499  37.020    53.480  37.009   
 53.462  36.999    53.444  36.989    53.427  36.978    53.409  36.968   
 53.391  36.958    53.374  36.948    53.356  36.938    53.339  36.928   
 53.322  36.918    53.304  36.908    53.287  36.898    53.270  36.888   
 53.253  36.878    53.236  36.869    53.220  36.859    53.203  36.849   
 53.186  36.840    53.170  36.830    53.153  36.821    53.137  36.811   
 53.121  36.802    53.105  36.793    53.089  36.783    53.073  36.774   
 53.057  36.765    53.041  36.756    53.025  36.747    53.009  36.738   
 52.994  36.729    52.978  36.720    52.963  36.711    52.948  36.702   
 52.932  36.693    52.917  36.684    52.902  36.675    52.887  36.667   
 52.872  36.658    52.857  36.650    52.842  36.641    52.828  36.633   
 52.813  36.624    52.798  36.616    52.784  36.607    52.769  36.599   
 52.755  36.591    52.741  36.582    52.727  36.574    52.713  36.566   
 52.698  36.558    52.684  36.550    52.671  36.542    52.657  36.534   
 52.643  36.526    52.629  36.518    52.616  36.510    52.602  36.502   
 52.589  36.495    52.575  36.487    52.562  36.479    52.549  36.471   
 52.535  36.464    52.522  36.456    52.509  36.449    52.496  36.441   
 52.483  36.434    52.470  36.426    52.458  36.419    52.445  36.412   
 52.432  36.404    52.420  36.397    52.407  36.390    52.395  36.383   
 52.382  36.375    52.370  36.368    52.358  36.361    52.345  36.354   
 52.333  36.347    52.321  36.340    52.309  36.333    52.297  36.326   
 52.285  36.319    52.273  36.313    52.262  36.306    52.250  36.299   
 52.238  36.292    52.227  36.286    52.215  36.279    52.204  36.272   
 52.192  36.266    52.181  36.259    52.170  36.253    52.158  36.246   
 52.147  36.240    52.136  36.233    52.125  36.227    52.114  36.221   
 52.103  36.214    52.092  36.208    52.081  36.202    52.071  36.195   
 52.060  36.189    52.049  36.183    52.039  36.177    52.028  36.171   
 52.017  36.165    52.007  36.159    51.997  36.153    51.986  36.147   
 51.976  36.141    51.966  36.135    51.956  36.129    51.945  36.123   
 51.935  36.117    51.925  36.112    51.915  36.106    51.905  36.100   
] scurve 
 50.000  54.000 moveto 
[  50.000  54.000    48.595  53.473    47.287  52.869    46.238  52.300   
 45.511  51.857    44.920  51.466    44.423  51.114    43.965  50.771   
 43.613  50.495    43.280  50.222    42.964  49.954    42.666  49.691   
 42.383  49.435    42.154  49.221    41.949  49.025    41.753  48.834   
 41.565  48.648    41.386  48.467    41.214  48.291    41.049  48.120   
 40.890  47.954    40.739  47.793    40.593  47.637    40.453  47.486   
 40.318  47.341    40.188  47.200    40.063  47.064    39.942  46.933   
 39.826  46.806    39.713  46.685    39.604  46.567    39.498  46.454   
 39.395  46.346    39.296  46.242    39.199  46.141    39.105  46.045   
 39.013  45.953    38.923  45.864    38.835  45.779    38.750  45.698   
 38.666  45.620    38.584  45.545    38.504  45.474    38.424  45.405   
 38.347  45.340    38.270  45.278    38.195  45.219    38.121  45.162   
 38.048  45.109    37.976  45.058    37.905  45.009    37.834  44.963   
 37.764  44.919    37.695  44.878    37.627  44.838    37.559  44.801   
 37.492  44.766    37.425  44.734    37.359  44.703    37.293  44.674   
 37.227  44.647    37.162  44.621    37.097  44.598    37.033  44.576   
 36.968  44.556    36.904  44.537    36.840  44.520    36.777  44.505   
 36.713  44.491    36.650  44.479    36.586  44.467    36.517  44.457   
 36.438  44.447    36.350  44.438    36.247  44.430    36.116  44.425   
 35.911  44.426    35.318  44.488    34.979  44.554    34.743  44.612   
 34.542  44.667    34.372  44.718    34.209  44.771    34.056  44.822   
 33.923  44.869    33.789  44.918    33.655  44.969    33.519  45.021   
 33.410  45.065    33.300  45.109    33.190  45.155    33.079  45.202   
 32.967  45.250    32.855  45.299    32.742  45.348    32.628  45.399   
 32.516  45.450    32.424  45.492    32.332  45.535    32.239  45.578   
 32.145  45.622    32.052  45.667    31.957  45.712    31.863  45.758   
 31.768  45.804    31.672  45.851    31.576  45.898    31.479  45.946   
 31.382  45.994    31.285  46.043    31.187  46.092    31.088  46.142   
 30.989  46.192    30.890  46.243    30.790  46.295    30.689  46.346   
 30.588  46.399    30.487  46.452    30.388  46.503    30.306  46.546   
 30.224  46.589    30.141  46.633    30.058  46.676    29.975  46.721   
 29.891  46.765    29.807  46.810    29.723  46.855    29.638  46.900   
 29.553  46.946    29.468  46.992    29.382  47.038    29.296  47.084   
 29.210  47.131    29.123  47.178    29.036  47.225    28.948  47.273   
 28.861  47.321    28.772  47.369    28.684  47.417    28.595  47.466   
 28.505  47.515    28.416  47.564    28.326  47.614    28.235  47.663   
 28.145  47.714    28.053  47.764    27.962  47.814    27.870  47.865   
 27.778  47.917    27.685  47.968    27.592  48.020    27.499  48.072   
 27.405  48.124    27.311  48.176    27.216  48.229    27.121  48.282   
 27.026  48.336    26.930  48.389    26.834  48.443    26.737  48.497   
 26.640  48.552    26.543  48.606    26.445  48.661    26.347  48.716   
 26.248  48.772    26.149  48.828    26.050  48.884    25.950  48.940   
 25.850  48.997    25.749  49.054    25.648  49.111    25.547  49.168   
 25.445  49.226    25.342  49.284    25.240  49.342    25.137  49.400   
 25.033  49.459    24.929  49.518    24.824  49.577    24.720  49.637   
 24.614  49.697    24.508  49.757    24.402  49.817    24.296  49.878   
 24.189  49.939    24.081  50.000    23.973  50.062    23.865  50.124   
 23.756  50.186    23.646  50.248   ] scurve 
[  23.646  50.248    23.537  50.311    23.426  50.374    23.316  50.437   
 23.205  50.500    23.093  50.564    22.981  50.628    22.868  50.692   
 22.755  50.757    22.642  50.822    22.528  50.887    22.414  50.953   
 22.299  51.018    22.183  51.085    22.074  51.147    21.981  51.201   
 21.888  51.254    21.794  51.308    21.700  51.362    21.606  51.416   
 21.511  51.470    21.416  51.524    21.321  51.579    21.226  51.634   
 21.130  51.688    21.034  51.744    20.937  51.799    20.840  51.855   
 20.743  51.910    20.646  51.966    20.548  52.022    20.450  52.079   
 20.352  52.135    20.253  52.192    20.154  52.249    20.054  52.306   
 19.955  52.363    19.855  52.421    19.754  52.478    19.654  52.536   
 19.552  52.594    19.451  52.653    19.349  52.711    19.247  52.770   
 19.145  52.829    19.042  52.888    18.939  52.947    18.836  53.007   
 18.732  53.066    18.628  53.126    18.523  53.186    18.418  53.247   
 18.313  53.307    18.208  53.368    18.102  53.429    17.996  53.490   
 17.889  53.551    17.782  53.613    17.675  53.675    17.567  53.737   
 17.459  53.799    17.351  53.861    17.242  53.924    17.133  53.987   
 17.024  54.050    16.914  54.113    16.804  54.176    16.693  54.240   
 16.582  54.304    16.471  54.368    16.359  54.432    16.247  54.497   
 16.135  54.562    16.022  54.627    15.909  54.692    15.796  54.757   
 15.682  54.823    15.567  54.889    15.453  54.955    15.338  55.021   
 15.222  55.088    15.107  55.154    14.990  55.221    14.874  55.289   
 14.757  55.356    14.640  55.424    14.522  55.492    14.404  55.560   
 14.285  55.628    14.166  55.697    14.047  55.765    13.927  55.834   
 13.807  55.904    13.687  55.973    13.566  56.043    13.444  56.113   
 13.323  56.183    13.201  56.254    13.078  56.324    12.955  56.395   
 12.832  56.466    12.708  56.538    12.584  56.609    12.459  56.681   
 12.334  56.753    12.209  56.826    12.083  56.898    11.957  56.971   
 11.830  57.044    11.703  57.117    11.576  57.191    11.448  57.265   
 11.319  57.339    11.191  57.413    11.061  57.488    10.932  57.563   
 10.802  57.638    10.671  57.713    10.540  57.789    10.409  57.864   
 10.277  57.940    10.145  58.017    10.012  58.093     9.879  58.170   
  9.745  58.247     9.611  58.325     9.477  58.402     9.342  58.480   
  9.206  58.558     9.071  58.637     8.934  58.715     8.798  58.794   
  8.660  58.873     8.523  58.953     8.385  59.033     8.246  59.113   
  8.107  59.193     7.968  59.273     7.828  59.354     7.687  59.435   
  7.546  59.516     7.405  59.598     7.263  59.680     7.121  59.762   
  6.978  59.845     6.835  59.927     6.691  60.010     6.547  60.093   
  6.402  60.177     6.257  60.261     6.112  60.345     5.966  60.429   
  5.819  60.514     5.672  60.599     5.524  60.684     5.376  60.769   
  5.228  60.855     5.079  60.941     4.929  61.028     4.779  61.114   
] scurve 
 50.000  63.000 moveto 
[  50.000  63.000    52.008  63.591    54.074  64.041    55.892  64.310   
 57.250  64.435    58.394  64.490    59.426  64.500    60.347  64.477   
 61.142  64.432    61.919  64.366    62.678  64.280    63.364  64.183   
 63.941  64.087    64.504  63.981    65.053  63.865    65.587  63.739   
 66.107  63.605    66.611  63.463    67.099  63.314    67.573  63.159   
 68.031  62.998    68.449  62.840    68.793  62.704    69.127  62.565   
 69.451  62.423    69.766  62.279    70.071  62.133    70.367  61.986   
 70.653  61.837    70.930  61.687    71.199  61.536    71.458  61.383   
 71.709  61.231    71.951  61.077    72.184  60.924    72.410  60.770   
 72.627  60.616    72.837  60.462    73.038  60.309    73.233  60.155   
 73.419  60.003    73.599  59.850    73.771  59.699    73.937  59.548   
 74.096  59.397    74.248  59.248    74.394  59.099    74.533  58.952   
 74.667  58.806    74.794  58.660    74.916  58.516    75.032  58.373   
 75.143  58.231    75.248  58.090    75.348  57.951    75.444  57.813   
 75.534  57.676    75.619  57.540    75.700  57.406    75.777  57.273   
 75.849  57.141    75.916  57.011    75.980  56.882    76.040  56.755   
 76.095  56.629    76.147  56.504    76.196  56.381    76.240  56.259   
 76.282  56.138    76.319  56.019    76.354  55.901    76.386  55.784   
 76.414  55.669    76.439  55.554    76.466  55.419    76.490  55.280   
 76.512  55.121    76.530  54.950    76.543  54.739    76.548  54.466   
 76.514  53.963    76.305  53.023    76.129  52.557    75.974  52.225   
 75.820  51.938    75.688  51.717    75.549  51.502    75.409  51.300   
 75.289  51.137    75.166  50.977    75.039  50.821    74.910  50.667   
 74.779  50.517    74.664  50.391    74.556  50.275    74.446  50.161   
 74.336  50.049    74.224  49.938    74.111  49.829    73.998  49.721   
 73.883  49.615    73.768  49.510    73.653  49.407    73.536  49.305   
 73.420  49.204    73.302  49.105    73.198  49.017    73.103  48.940   
 73.009  48.863    72.914  48.787    72.820  48.711    72.725  48.636   
 72.630  48.562    72.535  48.489    72.440  48.416    72.345  48.344   
 72.250  48.273    72.154  48.202    72.059  48.132    71.964  48.062   
 71.869  47.993    71.774  47.925    71.680  47.857    71.585  47.789   
 71.490  47.723    71.396  47.656    71.301  47.591    71.207  47.525   
 71.113  47.461    71.019  47.396    70.926  47.333    70.832  47.269   
 70.739  47.206    70.646  47.144    70.553  47.082    70.460  47.020   
 70.368  46.959    70.275  46.899    70.183  46.838    70.092  46.779   
 70.000  46.719    69.909  46.660    69.818  46.602    69.727  46.543   
 69.637  46.485    69.547  46.428    69.457  46.371    69.368  46.314   
 69.278  46.258    69.189  46.202    69.101  46.146    69.013  46.091   
 68.924  46.035    68.837  45.981    68.749  45.926    68.662  45.872   
 68.576  45.819    68.489  45.765    68.403  45.712    68.317  45.660   
 68.232  45.607    68.147  45.555    68.062  45.503    67.977  45.452   
 67.893  45.400    67.809  45.349    67.726  45.299    67.643  45.248   
 67.560  45.198    67.477  45.148    67.395  45.099    67.313  45.049   
 67.232  45.000    67.151  44.952    67.070  44.903    66.989  44.855   
 66.909  44.807    66.830  44.759    66.750  44.712    66.671  44.665   
 66.592  44.618    66.514  44.571    66.436  44.525    66.358  44.479   
 66.281  44.433    66.203  44.387    66.127  44.341    66.050  44.296   
 65.974  44.251    65.899  44.207   ] scurve 
[  65.899  44.207    65.823  44.162    65.748  44.118    65.673  44.074   
 65.599  44.030    65.525  43.986    65.451  43.943    65.378  43.900   
 65.305  43.857    65.232  43.814    65.160  43.771   ] scurve 
 50.000  63.000 moveto 
[  50.000  63.000    48.083  62.286    46.286  61.475    44.802  60.693   
 43.744  60.068    42.884  59.516    42.185  59.037    41.525  58.559   
 40.992  58.154    40.518  57.778    40.067  57.407    39.638  57.043   
 39.230  56.685    38.857  56.347    38.560  56.073    38.276  55.804   
 38.003  55.541    37.741  55.283    37.489  55.032    37.247  54.787   
 37.015  54.549    36.792  54.317    36.577  54.091    36.370  53.872   
 36.170  53.659    35.978  53.452    35.793  53.252    35.614  53.058   
 35.441  52.870    35.273  52.689    35.111  52.513    34.954  52.343   
 34.802  52.179    34.655  52.020    34.511  51.867    34.372  51.720   
 34.236  51.578    34.104  51.441    33.975  51.308    33.849  51.181   
 33.726  51.059    33.605  50.941    33.487  50.828    33.372  50.719   
 33.259  50.615    33.148  50.514    33.039  50.418    32.931  50.326   
 32.826  50.238    32.722  50.153    32.619  50.072    32.518  49.994   
 32.418  49.920    32.319  49.849    32.222  49.782    32.125  49.717   
 32.030  49.656    31.935  49.597    31.842  49.542    31.749  49.489   
 31.657  49.439    31.565  49.392    31.474  49.347    31.384  49.305   
 31.294  49.265    31.205  49.227    31.116  49.192    31.027  49.159   
 30.939  49.129    30.852  49.100    30.744  49.068    30.633  49.037   
 30.497  49.004    30.328  48.969    30.077  48.929    29.297  48.890   
 28.824  48.920    28.504  48.960    28.238  49.004    28.014  49.048   
 27.798  49.096    27.597  49.145    27.422  49.191    27.247  49.240   
 27.070  49.292    26.893  49.347    26.750  49.393    26.607  49.440   
 26.463  49.490    26.318  49.541    26.172  49.594    26.026  49.648   
 25.879  49.704    25.732  49.761    25.591  49.817    25.472  49.865   
 25.352  49.914    25.232  49.964    25.111  50.015    24.990  50.066   
 24.868  50.119    24.746  50.172    24.623  50.226    24.499  50.281   
 24.375  50.337    24.251  50.393    24.126  50.450    24.000  50.508   
 23.874  50.567    23.747  50.627    23.620  50.687    23.492  50.748   
 23.363  50.809    23.234  50.872    23.105  50.935    22.986  50.993   
 22.882  51.044    22.777  51.096    22.671  51.148    22.566  51.201   
 22.460  51.254    22.353  51.307    22.246  51.361    22.139  51.416   
 22.031  51.470    21.923  51.526    21.814  51.581    21.706  51.637   
 21.596  51.693    21.486  51.750    21.376  51.807    21.266  51.865   
 21.155  51.923    21.043  51.981    20.932  52.040    20.819  52.099   
 20.707  52.158    20.594  52.218    20.480  52.278    20.366  52.338   
 20.252  52.399    20.137  52.461    20.022  52.522    19.906  52.584   
 19.790  52.646    19.674  52.709    19.557  52.772    19.440  52.836   
 19.322  52.899    19.203  52.963    19.085  53.028    18.966  53.093   
 18.846  53.158    18.726  53.223    18.605  53.289    18.484  53.356   
 18.363  53.422    18.241  53.489    18.119  53.556    17.996  53.624   
 17.873  53.692    17.749  53.760    17.625  53.829    17.500  53.898   
 17.375  53.967    17.249  54.037    17.123  54.107    16.996  54.177   
 16.869  54.248    16.742  54.319    16.614  54.390    16.485  54.462   
 16.356  54.534    16.226  54.607    16.096  54.679    15.966  54.752   
 15.835  54.826    15.703  54.900    15.571  54.974    15.439  55.048   
 15.306  55.123    15.172  55.198    15.038  55.273    14.903  55.349   
 14.768  55.425    14.657  55.488    14.549  55.549    14.440  55.611   
 14.330  55.673    14.220  55.735   ] scurve 
grestore}
\LabelTeX   98.000  31.200 $x$\ELTX
\LabelTeX   46.500  68.000 $y$\ELTX
\LabelTeX   71.800  70.500 $\hbox{\rm isocline}~y=mx$\ELTX
\LabelTeX   71.800  66.500 $\hbox{\rm pente}~f(m)$\ELTX
\EndFig
\vskip6mm

\state Exemple -- |  L'\'equation $xy'(2y-x)=y^2$
peut se r\'ecrire
$$
y'=\frac{y^2}{x(2y-x)}\quad{\rm si}\quad x\not=0,\quad y\not=\frac{x}{2}.
$$
$y'$ est donc une fonction rationnelle en $x,y$ dont le num\'erateur et le d\'enominateur
sont des polyn\^omes homog\`enes de degr\'e 2. En divisant le num\'erateur et
d\'enominateur par $x^2$ on obtient
$$
y'=\frac{(y/x)^2}{2y/x-1}.
$$
Posons $z=\dfrac{y}{x}$, soit $y=xz$. Il vient
$$
\eqalign{
y'&=xz'+z=\frac{z^2}{2z-1},\cr
xz'&=\frac{z^2}{2z-1}-z=\frac{z-z^2}{2z-1}=\frac{z(1-z)}{2z-1}.
}
$$
$\bullet$ Solutions singuli\`eres~:
$$\eqalign{
z&=0,\qquad z=1,\cr
y&=0,\qquad y=x.\cr
}$$
\noindent$\bullet$ Pour $z\not=0$, $z\not=1$ l'\'equation se r\'ecrit
$$
\frac{2z-1}{z(1-z)}\ dz=\frac{dx}{x}.
$$
La fonction
$$
\frac{2z-1}{z(1-z)}=\frac{z-(1-z)}{z(1-z)}=\frac{1}{1-z}-\frac{1}{z}
$$

\noindent admet pour primitive
$$
-\ln\,|1-z|-\ln\,|z|=-\ln\,|z(1-z)|,
$$
d'o\`u le calcul des courbes int\'egrales~:
$$
\eqalign{
&\ln\,|z(1-z)|=-\ln\,|x|+C,\cr
&z(1-z)=\frac{\lambda}{x},\quad\frac{y}{x}\,\Big(1-\frac{y}{x}\Big)=\frac{\lambda}{x},\cr
&y(x-y)=\lambda x.\cr
}
$$
Les courbes int\'egrales sont donc des coniques. On peut mettre l'\'equation sous la forme
$$
(y-\lambda)(x-y-\lambda)=\lambda^2
$$
c'est-\`a-dire $XY=\lambda$ avec $X=x-y-\lambda$ et $Y=y-\lambda$. Il s'agit d'une
hyperbole d'asymptotes $y=\lambda$, $y=x-\lambda$ (parall\`eles aux directions
asymptotiques $y=0$, $y=x$ donn\'ees par les droites int\'egrales singuli\`eres).

\bigskip
\majedeb
\state Exercice -- |  {\it Montrer que chaque hyperbole passe par $(0,0)$ avec
tangente $x=0$.} 
\par\majefin\par

\InsertFig  9.000 79.000 {
1 mm unit
  0.000  35.000 moveto 
100.000   0.000 vector 
 50.000   0.000 moveto 
 70.000  90.000 vector 
gsave 
  5.000   5.000 moveto 
 90.000  60.000 rect  clip  newpath 
 20.000   5.000 moveto
 80.000  65.000 lineto
 45.720   5.010 moveto 
[  45.720   5.010    45.397   3.809    45.120   2.811   ] scurve 
 45.720   5.010 moveto 
[  45.720   5.010    46.011   6.124    46.216   6.931    46.406   7.694   
 46.582   8.417    46.745   9.102    46.896   9.753    47.038  10.372   
 47.170  10.962    47.293  11.524    47.408  12.060    47.517  12.572   
 47.618  13.062    47.713  13.531    47.803  13.980    47.888  14.411   
 47.968  14.825    48.043  15.222    48.114  15.604    48.182  15.971   
 48.245  16.325    48.306  16.666    48.363  16.994    48.418  17.312   
 48.470  17.618    48.519  17.913    48.566  18.199    48.611  18.475   
 48.654  18.743    48.695  19.002    48.734  19.253    48.771  19.496   
 48.807  19.731    48.841  19.960    48.873  20.182    48.905  20.397   
 48.935  20.606    48.963  20.810    48.991  21.007    49.018  21.200   
 49.043  21.387    49.068  21.569    49.091  21.746    49.114  21.918   
 49.136  22.087    49.157  22.250    49.177  22.410    49.197  22.566   
 49.216  22.718    49.234  22.866    49.252  23.011    49.269  23.153   
 49.285  23.291    49.301  23.426    49.316  23.557    49.331  23.686   
 49.346  23.812    49.359  23.935    49.373  24.056    49.386  24.174   
 49.399  24.289    49.411  24.402    49.423  24.513    49.434  24.621   
 49.446  24.727    49.456  24.831    49.467  24.933    49.477  25.033   
 49.487  25.131    49.497  25.227    49.506  25.321    49.515  25.413   
 49.524  25.504    49.533  25.593    49.541  25.680    49.550  25.766   
 49.558  25.850    49.565  25.933    49.573  26.014    49.580  26.093   
 49.587  26.172    49.594  26.249    49.601  26.324    49.608  26.398   
 49.614  26.471    49.621  26.543    49.627  26.614    49.633  26.683   
 49.639  26.751    49.644  26.819    49.650  26.885    49.655  26.950   
 49.661  27.014    49.666  27.077    49.671  27.139    49.676  27.200   
 49.681  27.260    49.686  27.319    49.690  27.377    49.695  27.435   
 49.699  27.491    49.704  27.547    49.708  27.602    49.712  27.656   
 49.716  27.709    49.720  27.762    49.724  27.813    49.728  27.864   
 49.732  27.915    49.735  27.964    49.739  28.013    49.743  28.062   
 49.746  28.109    49.749  28.156    49.753  28.202    49.756  28.248   
 49.759  28.293    49.762  28.338    49.765  28.381    49.768  28.425   
 49.771  28.468    49.774  28.510    49.777  28.551    49.780  28.593   
 49.783  28.633    49.785  28.673    49.788  28.713    49.790  28.752   
 49.793  28.791    49.795  28.829    49.798  28.866    49.800  28.904   
 49.803  28.944    49.806  28.989    49.809  29.034    49.811  29.078   
 49.814  29.121    49.817  29.164    49.819  29.206    49.822  29.247   
 49.824  29.288    49.827  29.328    49.829  29.368    49.831  29.407   
 49.834  29.446    49.836  29.484    49.838  29.522    49.840  29.559   
 49.842  29.595    49.845  29.631    49.847  29.667    49.849  29.702   
 49.851  29.737    49.852  29.771    49.854  29.805    49.856  29.838   
 49.858  29.871    49.860  29.904    49.861  29.936    49.863  29.967   
 49.865  29.999    49.866  30.030    49.868  30.060    50.000  35.000
] scurve 
 30.000  25.000 moveto 
[  30.000  25.000    29.744  23.709    29.661  23.497    29.574  23.293   
 29.476  23.081    29.367  22.861    29.247  22.633    29.116  22.396   
 28.973  22.150    28.817  21.894    28.649  21.628    28.467  21.350   
 28.272  21.061    28.061  20.760    27.835  20.445    27.593  20.116   
 27.334  19.771    27.056  19.411    26.760  19.032    26.442  18.635   
 26.103  18.218    25.740  17.779    25.351  17.316    24.828  16.702   
 24.258  16.043    23.636  15.335    22.957  14.572    22.215  13.748   
 21.401  12.854    20.507  11.881    19.521  10.819    18.432   9.655   
 17.223   8.374    15.876   6.955    14.367   5.378    12.669   3.612   
] scurve 
 30.000  25.000 moveto 
[  30.000  25.000    29.746  26.036    29.692  26.121    29.659  26.170   
 29.625  26.219    29.588  26.266    29.550  26.314    29.510  26.360   
 29.468  26.406    29.424  26.452    29.379  26.497    29.332  26.542   
 29.282  26.587    29.231  26.630    29.178  26.674    29.123  26.717   
 29.066  26.759    29.007  26.801    28.946  26.843    28.883  26.884   
 28.818  26.925    28.751  26.965    28.681  27.005    28.609  27.045   
 28.535  27.084    28.459  27.123    28.380  27.161    28.299  27.200   
 28.215  27.237    28.129  27.275    28.041  27.312    27.950  27.348   
 27.856  27.385    27.759  27.420    27.660  27.456    27.557  27.491   
 27.452  27.526    27.344  27.561    27.233  27.595    27.119  27.629   
 27.001  27.663    26.881  27.696    26.756  27.729    26.629  27.762   
 26.498  27.795    26.363  27.827    26.225  27.859    26.082  27.890   
 25.936  27.922    25.786  27.953    25.631  27.984    25.472  28.014   
 25.309  28.044    25.141  28.074    24.968  28.104    24.791  28.134   
 24.608  28.163    24.421  28.192    24.227  28.221    24.029  28.249   
 23.824  28.277    23.614  28.305    23.398  28.333    23.175  28.361   
 22.946  28.388    22.709  28.415    22.466  28.442    22.215  28.469   
 21.957  28.495    21.691  28.522    21.416  28.548    21.133  28.574   
 20.873  28.596    20.634  28.617    20.388  28.637    20.136  28.657   
 19.877  28.677    19.612  28.697    19.339  28.717    19.059  28.736   
 18.772  28.756    18.477  28.775    18.173  28.795    17.862  28.814   
 17.541  28.833    17.211  28.852    16.872  28.870    16.523  28.889   
 16.164  28.908    15.794  28.926    15.413  28.945    15.020  28.963   
 14.615  28.981    14.197  28.999    13.767  29.017    13.322  29.035   
 12.863  29.052    12.389  29.070    11.899  29.087    11.439  29.103   
 11.022  29.117    10.594  29.131    10.154  29.145     9.701  29.158   
  9.235  29.172     8.756  29.185     8.263  29.199     7.754  29.212   
  7.231  29.226     6.691  29.239     6.134  29.252     5.560  29.265   
  4.967  29.278     4.355  29.292   ] scurve 
 15.000  20.000 moveto 
[  15.000  20.000    15.416  19.327    15.460  19.226    15.498  19.131   
 15.532  19.035    15.564  18.938    15.593  18.839    15.619  18.739   
 15.642  18.638    15.662  18.536    15.679  18.432    15.693  18.327   
 15.704  18.221    15.712  18.113    15.717  18.004    15.720  17.894   
 15.719  17.782    15.716  17.669    15.709  17.554    15.700  17.438   
 15.688  17.320    15.673  17.200    15.655  17.079    15.634  16.956   
 15.610  16.832    15.583  16.705    15.553  16.577    15.520  16.448   
 15.485  16.316    15.446  16.182    15.404  16.047    15.359  15.909   
 15.311  15.770    15.259  15.628    15.205  15.485    15.147  15.339   
 15.087  15.191    15.023  15.041    14.955  14.889    14.885  14.734   
 14.811  14.577    14.734  14.417    14.653  14.255    14.569  14.090   
 14.481  13.922    14.390  13.752    14.295  13.579    14.196  13.404   
 14.094  13.225    13.988  13.043    13.878  12.859    13.740  12.632   
 13.591  12.392    13.435  12.147    13.273  11.896    13.105  11.640   
 12.929  11.378    12.725  11.079    12.486  10.735    12.235  10.380   
 11.972  10.015    11.696   9.639    11.393   9.233    11.012   8.731   
 10.609   8.209    10.183   7.666     9.731   7.100     9.165   6.400   
  8.523   5.621     7.834   4.798     7.095   3.928   ] scurve 
 15.000  20.000 moveto 
[  15.000  20.000    14.406  20.621    14.305  20.705    14.207  20.783   
 14.105  20.859    13.999  20.935    13.890  21.009    13.777  21.083   
 13.659  21.157    13.538  21.229    13.412  21.301    13.282  21.372   
 13.148  21.442    13.010  21.511    12.867  21.580    12.720  21.648   
 12.568  21.716    12.411  21.783    12.249  21.849    12.083  21.914   
 11.911  21.979    11.734  22.043    11.552  22.106    11.365  22.169   
 11.171  22.232    10.973  22.293    10.768  22.354    10.557  22.415   
 10.340  22.475    10.117  22.534     9.888  22.593     9.651  22.651   
  9.408  22.709     9.158  22.766     8.900  22.823     8.635  22.879   
  8.362  22.934     8.081  22.989     7.792  23.044     7.495  23.098   
  7.189  23.152     6.873  23.205     6.549  23.258     6.215  23.310   
  5.870  23.362     5.516  23.413     5.150  23.464     4.774  23.514   
] scurve 
 70.000  45.000 moveto 
[  70.000  45.000    70.254  43.964    70.308  43.879    70.341  43.830   
 70.375  43.781    70.412  43.734    70.450  43.686    70.490  43.640   
 70.532  43.594    70.576  43.548    70.621  43.503    70.668  43.458   
 70.718  43.413    70.769  43.370    70.822  43.326    70.877  43.283   
 70.934  43.241    70.993  43.199    71.054  43.157    71.117  43.116   
 71.182  43.075    71.249  43.035    71.319  42.995    71.391  42.955   
 71.465  42.916    71.541  42.877    71.620  42.839    71.701  42.800   
 71.785  42.763    71.871  42.725    71.959  42.688    72.050  42.652   
 72.144  42.615    72.241  42.580    72.340  42.544    72.443  42.509   
 72.548  42.474    72.656  42.439    72.767  42.405    72.881  42.371   
 72.999  42.337    73.119  42.304    73.244  42.271    73.371  42.238   
 73.502  42.205    73.637  42.173    73.775  42.141    73.918  42.110   
 74.064  42.078    74.214  42.047    74.369  42.016    74.528  41.986   
 74.691  41.956    74.859  41.926    75.032  41.896    75.209  41.866   
 75.392  41.837    75.579  41.808    75.773  41.779    75.971  41.751   
 76.176  41.723    76.386  41.695    76.602  41.667    76.825  41.639   
 77.054  41.612    77.291  41.585    77.534  41.558    77.785  41.531   
 78.043  41.505    78.309  41.478    78.584  41.452    78.867  41.426   
 79.127  41.404    79.366  41.383    79.612  41.363    79.864  41.343   
 80.123  41.323    80.388  41.303    80.661  41.283    80.941  41.264   
 81.228  41.244    81.523  41.225    81.827  41.205    82.138  41.186   
 82.459  41.167    82.789  41.148    83.128  41.130    83.477  41.111   
 83.836  41.092    84.206  41.074    84.587  41.055    84.980  41.037   
 85.385  41.019    85.803  41.001    86.233  40.983    86.678  40.965   
 87.137  40.948    87.611  40.930    88.101  40.913    88.561  40.897   
 88.978  40.883    89.406  40.869    89.846  40.855    90.299  40.842   
 90.765  40.828    91.244  40.815    91.737  40.801    92.246  40.788   
 92.769  40.774    93.309  40.761    93.866  40.748    94.440  40.735   
 95.033  40.722    95.645  40.708   ] scurve 
 70.000  45.000 moveto 
[  70.000  45.000    70.256  46.291    70.339  46.503    70.426  46.707   
 70.524  46.919    70.633  47.139    70.753  47.367    70.884  47.604   
 71.027  47.850    71.183  48.106    71.351  48.372    71.533  48.650   
 71.728  48.939    71.939  49.240    72.165  49.555    72.407  49.884   
 72.666  50.229    72.944  50.589    73.240  50.968    73.558  51.365   
 73.897  51.782    74.260  52.221    74.649  52.684    75.172  53.298   
 75.742  53.957    76.364  54.665    77.043  55.428    77.785  56.252   
 78.599  57.146    79.493  58.119    80.479  59.181    81.568  60.345   
 82.777  61.626    84.124  63.045    85.633  64.622    87.331  66.388   
] scurve 
 84.000  52.000 moveto 
[  84.000  52.000    84.149  50.921    84.189  50.798    84.228  50.694   
 84.271  50.591    84.317  50.489    84.367  50.389    84.421  50.290   
 84.478  50.192    84.540  50.096    84.605  50.000    84.674  49.906   
 84.746  49.813    84.823  49.722    84.904  49.631    84.989  49.541   
 85.078  49.453    85.171  49.366    85.268  49.279    85.370  49.194   
 85.476  49.110    85.586  49.026    85.701  48.944    85.820  48.863   
 85.945  48.782    86.074  48.703    86.208  48.624    86.346  48.547   
 86.490  48.470    86.639  48.394    86.794  48.319    86.954  48.245   
 87.119  48.171    87.290  48.099    87.467  48.027    87.650  47.956   
 87.838  47.885    88.034  47.816    88.235  47.747    88.443  47.679   
 88.658  47.612    88.880  47.545    89.109  47.479    89.345  47.414   
 89.589  47.350    89.841  47.286    90.101  47.223    90.324  47.170   
 90.544  47.120    90.769  47.071    91.000  47.022    91.237  46.974   
 91.480  46.926    91.729  46.878    91.984  46.831    92.246  46.784   
 92.515  46.737    92.791  46.691    93.074  46.645    93.364  46.600   
 93.662  46.555    93.968  46.510    94.282  46.465    94.604  46.421   
 94.935  46.377    95.274  46.334   ] scurve 
 84.000  52.000 moveto 
[  84.000  52.000    84.149  53.228    84.207  53.454    84.272  53.681   
 84.346  53.913    84.429  54.151    84.521  54.395    84.622  54.645   
 84.733  54.902    84.854  55.166    84.984  55.437    85.124  55.715   
 85.275  56.001    85.436  56.295    85.608  56.597    85.791  56.908   
 85.986  57.228    86.192  57.557    86.410  57.897    86.641  58.246   
 86.885  58.607    87.142  58.979    87.413  59.363    87.698  59.759   
 87.998  60.169    88.314  60.592    88.647  61.030    89.012  61.504   
 89.475  62.094    89.967  62.711    90.490  63.356    91.060  64.048   
 91.806  64.941    92.613  65.891   ] scurve 
 55.000  25.000 moveto 
[  55.000  25.000    55.704  24.465    55.911  24.317    56.101  24.184   
 56.284  24.059    56.475  23.931    56.675  23.801    56.884  23.667   
 57.104  23.530    57.335  23.389    57.577  23.246    57.832  23.098   
 58.100  22.947    58.383  22.792    58.664  22.641    58.914  22.511   
 59.174  22.378    59.447  22.242    59.732  22.104    60.030  21.962   
 60.343  21.817    60.672  21.669    61.016  21.517    61.378  21.362   
 61.759  21.203    62.160  21.041    62.504  20.905    62.854  20.770   
 63.221  20.632    63.604  20.491    64.004  20.348    64.423  20.202   
 64.862  20.053    65.322  19.900    65.804  19.745    66.310  19.586   
 66.843  19.424    67.340  19.277    67.806  19.143    68.293  19.006   
 68.800  18.867    69.330  18.725    69.884  18.581    70.463  18.434   
 71.069  18.285    71.704  18.132    72.370  17.978    73.068  17.820   
 73.760  17.668    74.373  17.537    75.010  17.405    75.674  17.270   
 76.367  17.133    77.090  16.994    77.844  16.853    78.633  16.710   
 79.459  16.565    80.323  16.417    81.228  16.267    82.177  16.114   
 82.989  15.987    83.815  15.861    84.675  15.734    85.571  15.605   
 86.504  15.474    87.478  15.342    88.495  15.208    89.557  15.071   
 90.667  14.933    91.829  14.794    93.046  14.652    94.157  14.526   
 95.216  14.410   ] scurve 
 55.000  25.000 moveto 
[  55.000  25.000    54.418  25.488    54.284  25.606    54.157  25.721   
 54.035  25.833    53.919  25.943    53.807  26.050    53.700  26.154   
 53.598  26.256    53.500  26.356    53.406  26.454    53.316  26.549   
 53.229  26.642    53.146  26.733    53.066  26.823    52.989  26.910   
 52.914  26.995    52.843  27.079    52.774  27.161    52.708  27.241   
 52.644  27.320    52.582  27.397    52.523  27.473    52.465  27.547   
 52.410  27.619    52.356  27.690    52.304  27.760    52.254  27.829   
 52.205  27.896    52.158  27.962    52.113  28.027    52.069  28.090   
 52.026  28.153    51.985  28.214    51.945  28.274    51.906  28.334   
 51.868  28.392    51.831  28.449    51.796  28.505    51.761  28.560   
 51.728  28.614    51.695  28.668    51.664  28.720    51.633  28.772   
 51.603  28.823    51.574  28.873    51.545  28.922    51.518  28.970   
 51.491  29.018    51.465  29.065    51.439  29.111    51.415  29.156   
 51.391  29.201    51.367  29.245    51.344  29.288    51.322  29.331   
 51.300  29.373    51.279  29.415    51.258  29.455    51.238  29.496   
 51.218  29.535    51.199  29.574    51.180  29.613    51.161  29.651   
 51.143  29.689    51.126  29.726    51.109  29.762    51.092  29.798   
 51.076  29.833    51.060  29.868    51.044  29.903    51.029  29.937   
 51.014  29.971    50.999  30.004    50.985  30.036    50.971  30.069   
 50.957  30.101    50.944  30.132    50.930  30.163    50.918  30.194   
 50.905  30.224    50.893  30.254    50.880  30.284    50.869  30.313   
 50.857  30.342    50.846  30.370    50.834  30.398    50.823  30.426   
 50.813  30.453    50.802  30.480    50.792  30.507    50.782  30.534   
 50.772  30.560    50.762  30.586    50.753  30.611    50.743  30.637   
 50.734  30.662    50.725  30.686    50.716  30.711    50.707  30.735   
 50.699  30.759    50.690  30.782    50.682  30.806    50.674  30.829   
 50.666  30.851    50.658  30.874    50.651  30.896    50.643  30.919   
 50.636  30.940    50.629  30.962    50.621  30.983    50.614  31.005   
 50.607  31.026    50.601  31.046    50.594  31.067    50.587  31.087   
 50.581  31.107    50.575  31.127    50.568  31.147    50.562  31.166   
 50.556  31.186    50.550  31.205    50.544  31.224    50.538  31.242   
 50.533  31.261    50.527  31.279    50.522  31.297    50.516  31.315   
 50.511  31.333    50.505  31.352    50.499  31.374    50.493  31.395   
 50.486  31.417    50.480  31.438    50.474  31.459    50.469  31.479   
 50.463  31.499    50.457  31.520    50.452  31.539    50.446  31.559   
 50.441  31.578    50.436  31.598    50.430  31.617    50.425  31.635   
 50.420  31.654    50.415  31.672    50.411  31.691    50.406  31.708   
 50.401  31.726    50.397  31.744    50.392  31.761    50.388  31.778   
 50.384  31.795    50.379  31.812    50.375  31.829    50.371  31.845   
 50.367  31.862    50.363  31.878    50.359  31.894    50.355  31.910   
 50.351  31.925    50.348  31.941    50.344  31.956    50.340  31.972   
 50.337  31.987    50.333  32.001    50.329  32.020    50.325  32.038   
 50.321  32.056    50.317  32.073    50.313  32.091    50.309  32.108   
 50.305  32.126    50.301  32.142    50.297  32.159    50.294  32.176   
 50.290  32.192    50.287  32.208    50.283  32.224    50.280  32.240   
 50.277  32.256    50.273  32.271    50.270  32.286    50.267  32.301   
 50.264  32.316    50.261  32.331    50.258  32.346    50.255  32.360   
 50.252  32.374    50.000  35.000   ] scurve 
[  50.249  32.389    50.246  32.403    50.244  32.416    50.241  32.430   
 50.238  32.444    50.236  32.457    50.233  32.470    50.231  32.483   
 50.228  32.496    50.226  32.509    50.223  32.522    50.221  32.535   
 50.219  32.547   ] scurve 
 60.000  30.000 moveto 
[  60.000  30.000    61.109  29.870    61.430  29.836    61.728  29.804   
 62.007  29.776    62.297  29.747    62.599  29.717    62.913  29.688   
 63.240  29.658    63.582  29.628    63.938  29.597    64.310  29.566   
 64.698  29.535    65.105  29.503    65.506  29.473    65.861  29.447   
 66.228  29.421    66.611  29.395    67.009  29.368    67.423  29.341   
 67.854  29.314    68.304  29.287    68.773  29.259    69.263  29.231   
 69.774  29.203    70.310  29.175    70.767  29.151    71.230  29.128   
 71.712  29.105    72.213  29.081    72.734  29.058    73.276  29.034   
 73.840  29.010    74.429  28.986    75.043  28.961    75.685  28.936   
 76.355  28.912    76.978  28.889    77.560  28.869    78.164  28.849   
 78.791  28.828    79.444  28.808    80.122  28.787    80.828  28.766   
 81.564  28.745    82.332  28.724    83.132  28.703    83.969  28.681   
 84.794  28.661    85.521  28.644    86.275  28.626    87.058  28.609   
 87.872  28.591    88.717  28.573    89.597  28.556    90.514  28.538   
 91.469  28.520    92.465  28.501    93.504  28.483    94.591  28.465   
 95.496  28.450   ] scurve 
 60.000  30.000 moveto 
[  60.000  30.000    59.056  30.126    58.835  30.157    58.623  30.188   
 58.420  30.218    58.224  30.248    58.035  30.278    57.853  30.307   
 57.678  30.336    57.508  30.364    57.345  30.393    57.187  30.420   
 57.035  30.448    56.887  30.475    56.745  30.502    56.607  30.529   
 56.474  30.555    56.345  30.581    56.220  30.606    56.098  30.632   
 55.981  30.657    55.867  30.682    55.756  30.706    55.649  30.730   
 55.545  30.754    55.443  30.778    55.345  30.801    55.250  30.824   
 55.157  30.847    55.066  30.870    54.978  30.892    54.893  30.915   
 54.810  30.936    54.729  30.958    54.650  30.980    54.573  31.001   
 54.498  31.022    54.425  31.043    54.354  31.063    54.285  31.084   
 54.217  31.104    54.151  31.124    54.087  31.143    54.024  31.163   
 53.962  31.182    53.902  31.201    53.844  31.220    53.787  31.239   
 53.731  31.258    53.676  31.276    53.623  31.294    53.571  31.312   
 53.520  31.330    53.470  31.348    53.421  31.365    53.373  31.383   
 53.327  31.400    53.281  31.417    53.236  31.434    53.192  31.450   
 53.150  31.467    53.108  31.483    53.066  31.500    53.026  31.516   
 52.987  31.532    52.948  31.548    52.910  31.563    52.873  31.579   
 52.836  31.594    52.801  31.609    52.766  31.624    52.731  31.639   
 52.697  31.654    52.664  31.669    52.632  31.684    52.600  31.698   
 52.569  31.712    52.538  31.726    52.508  31.741    52.478  31.755   
 52.449  31.768    52.421  31.782    52.393  31.796    52.365  31.809   
 52.338  31.823    52.311  31.836    52.285  31.849    52.260  31.862   
 52.234  31.875    52.210  31.888    52.185  31.901    52.161  31.913   
 52.138  31.926    52.115  31.938    52.092  31.950    52.069  31.963   
 52.047  31.975    52.026  31.987    52.004  31.999    51.983  32.011   
 51.963  32.022    51.942  32.034    51.922  32.046    51.903  32.057   
 51.883  32.068    51.864  32.080    51.845  32.091    51.827  32.102   
 51.808  32.113    51.791  32.124    51.773  32.135    51.755  32.146   
 51.738  32.156    51.721  32.167    51.705  32.178    51.688  32.188   
 51.672  32.198    51.656  32.209    51.641  32.219    51.625  32.229   
 51.610  32.239    51.595  32.249    51.580  32.259    51.565  32.269   
 51.551  32.279    51.537  32.289    51.523  32.298    51.509  32.308   
 51.495  32.317    51.481  32.328    51.464  32.339    51.448  32.351   
 51.432  32.362    51.416  32.374    51.401  32.385    51.386  32.396   
 51.371  32.408    51.356  32.419    51.342  32.430    51.327  32.440   
 51.313  32.451    51.300  32.462    51.286  32.472    51.273  32.483   
 51.260  32.493    51.247  32.504    51.234  32.514    51.221  32.524   
 51.209  32.534    51.197  32.544    51.185  32.554    51.173  32.564   
 51.162  32.574    51.150  32.583    51.139  32.593    51.128  32.602   
 51.117  32.612    51.106  32.621    51.095  32.630    51.085  32.640   
 51.075  32.649    51.065  32.658    51.055  32.667    51.045  32.676   
 51.035  32.685    51.025  32.694    51.014  32.704    51.002  32.715   
 50.991  32.726    50.980  32.736    50.969  32.747    50.958  32.757   
 50.947  32.767    50.937  32.778    50.926  32.788    50.916  32.798   
 50.906  32.808    50.896  32.818    50.887  32.827    50.877  32.837   
 50.868  32.847    50.859  32.856    50.850  32.866    50.841  32.875   
 50.832  32.884    50.824  32.894    50.815  32.903    50.807  32.912   
 50.798  32.921    50.000  35.000   ] scurve 
[  50.790  32.929    50.782  32.938    50.775  32.947    50.767  32.956   
 50.759  32.964    50.752  32.973    50.744  32.981    50.737  32.990   
 50.730  32.998    50.723  33.006    50.716  33.014    50.709  33.022   
 50.702  33.030    50.695  33.039    50.687  33.049    50.679  33.059   
 50.671  33.068    50.663  33.078    50.655  33.087    50.648  33.097   
 50.640  33.106    50.633  33.115    50.626  33.124    50.619  33.133   
 50.612  33.142   ] scurve 
 61.000  10.000 moveto 
[  61.000  10.000    61.680   9.380    62.102   9.008    62.548   8.626   
 62.984   8.260    63.378   7.937    63.792   7.605    64.225   7.266   
 64.679   6.918    65.155   6.561    65.655   6.195    66.181   5.819   
 66.734   5.434    67.317   5.038    67.931   4.631   ] scurve 
 61.000  10.000 moveto 
[  61.000  10.000    60.377  10.593    60.042  10.922    59.723  11.242   
 59.419  11.553    59.129  11.857    58.853  12.153    58.589  12.441   
 58.337  12.722    58.096  12.997    57.865  13.264    57.644  13.525   
 57.432  13.780    57.229  14.029    57.034  14.272    56.847  14.510   
 56.668  14.742    56.495  14.969    56.329  15.191    56.170  15.408   
 56.016  15.621    55.869  15.829    55.726  16.032    55.589  16.231   
 55.456  16.426    55.329  16.617    55.205  16.804    55.086  16.988   
 54.972  17.167    54.860  17.343    54.753  17.516    54.649  17.685   
 54.549  17.852    54.452  18.014    54.357  18.174    54.266  18.331   
 54.178  18.485    54.092  18.636    54.009  18.785    53.929  18.931   
 53.851  19.074    53.775  19.214    53.702  19.352    53.630  19.488   
 53.561  19.622    53.494  19.753    53.428  19.882    53.364  20.008   
 53.303  20.133    53.242  20.256    53.184  20.376    53.127  20.495   
 53.071  20.612    53.017  20.727    52.965  20.840    52.914  20.951   
 52.864  21.060    52.815  21.168    52.768  21.274    52.722  21.379   
 52.677  21.482    52.633  21.583    52.590  21.683    52.548  21.782   
 52.507  21.879    52.467  21.974    52.429  22.069    52.391  22.161   
 52.354  22.253    52.317  22.343    52.282  22.432    52.247  22.520   
 52.214  22.606    52.180  22.692    52.148  22.776    52.117  22.859   
 52.086  22.941    52.055  23.021    52.026  23.101    51.997  23.180   
 51.969  23.257    51.941  23.334    51.914  23.410    51.887  23.484   
 51.861  23.558    51.836  23.631    51.811  23.703    51.786  23.774   
 51.762  23.844    51.739  23.913    51.716  23.981    51.693  24.049   
 51.671  24.115    51.649  24.181    51.628  24.246    51.607  24.310   
 51.587  24.374    51.567  24.437    51.547  24.499    51.528  24.560   
 51.509  24.621    51.490  24.681    51.472  24.740    51.454  24.798   
 51.436  24.856    51.419  24.913    51.402  24.970    51.385  25.026   
 51.369  25.081    51.353  25.136    51.337  25.190    51.322  25.244   
 51.306  25.297    51.291  25.349    51.277  25.401    51.262  25.452   
 51.248  25.503    51.234  25.553    51.220  25.603    51.207  25.652   
 51.193  25.700    51.180  25.748    51.167  25.796    51.155  25.843   
 51.142  25.890    51.130  25.936    51.118  25.982    51.106  26.027   
 51.094  26.072    51.083  26.116    51.071  26.160    51.060  26.203   
 51.049  26.249    51.035  26.303    51.022  26.355    51.009  26.407   
 50.996  26.458    50.984  26.509    50.971  26.559    50.959  26.609   
 50.948  26.658    50.936  26.706    50.925  26.754    50.914  26.801   
 50.903  26.848    50.892  26.894    50.881  26.940    50.871  26.985   
 50.861  27.030    50.851  27.074    50.841  27.117    50.831  27.161   
 50.822  27.203    50.812  27.246    50.803  27.287    50.794  27.329   
 50.785  27.370    50.777  27.410    50.768  27.450    50.759  27.490   
 50.751  27.529    50.743  27.568    50.735  27.606    50.727  27.644   
 50.719  27.682    50.712  27.719    50.704  27.756    50.697  27.792   
 50.689  27.829    50.681  27.872    50.672  27.917    50.663  27.960   
 50.655  28.003    50.646  28.046    50.638  28.088    50.630  28.129   
 50.622  28.170    50.615  28.211    50.607  28.251    50.600  28.290   
 50.592  28.329    50.585  28.368    50.578  28.406    50.571  28.444   
 50.565  28.481    50.558  28.518    50.551  28.555    50.545  28.591   
 50.539  28.627    50.533  28.662   ] scurve 
[  50.533  28.662    50.395  29.514    50.380  29.616    50.367  29.708   
 50.354  29.797    50.342  29.883    50.331  29.966    50.320  30.046   
 50.310  30.124    50.300  30.199    50.291  30.272    50.282  30.343   
 50.273  30.412    50.265  30.479    50.257  30.544    50.250  30.607   
 50.243  30.668    50.236  30.727    50.230  30.785    50.223  30.842   
 50.217  30.897    50.211  30.950    50.206  31.002    50.201  31.053   
 50.195  31.103    50.191  31.151    50.186  31.198    50.181  31.244   
 50.177  31.289    50.173  31.333    50.168  31.376    50.165  31.418   
 50.161  31.458    50.157  31.498    50.153  31.537    50.150  31.576   
 50.147  31.613    50.143  31.649    50.140  31.685    50.137  31.720   
 50.134  31.754    50.132  31.788    50.129  31.821    50.126  31.853
 50.000  35.000   ] scurve 
  6.440  55.710 moveto 
[   6.440  55.710     7.873  55.540     9.048  55.397    10.169  55.255   
 11.242  55.115    12.269  54.977    13.253  54.841    14.195  54.707   
 15.100  54.574    15.968  54.444    16.801  54.315    17.603  54.188   
 18.374  54.062    19.116  53.938    19.830  53.816    20.518  53.695   
 21.182  53.576    21.822  53.459    22.440  53.343    23.036  53.228   
 23.612  53.115    24.169  53.003    24.707  52.892    25.228  52.783   
 25.732  52.675    26.220  52.569    26.693  52.464    27.151  52.360   
 27.594  52.257    28.025  52.155    28.442  52.055    28.848  51.956   
 29.241  51.858    29.623  51.761    29.994  51.665    30.354  51.571   
 30.705  51.477    31.045  51.384    31.377  51.293    31.699  51.202   
 32.013  51.113    32.318  51.024    32.616  50.937    32.905  50.850   
 33.188  50.764    33.463  50.680    33.731  50.596    33.992  50.513   
 34.247  50.431    34.495  50.350    34.738  50.269    34.974  50.190   
 35.205  50.111    35.431  50.033    35.651  49.956    35.866  49.880   
 36.076  49.804    36.282  49.730    36.482  49.656    36.679  49.582   
 36.870  49.510    37.058  49.438    37.241  49.367    37.421  49.296   
 37.596  49.227    37.768  49.158    37.936  49.089    38.101  49.022   
 38.262  48.954    38.420  48.888    38.574  48.822    38.726  48.757   
 38.874  48.692    39.019  48.629    39.162  48.565    39.302  48.502   
 39.438  48.440    39.573  48.379    39.704  48.317    39.833  48.257   
 39.960  48.197    40.084  48.138    40.206  48.079    40.325  48.020   
 40.443  47.962    40.558  47.905    40.671  47.848    40.782  47.792   
 40.891  47.736    40.998  47.681    41.103  47.626    41.206  47.571   
 41.307  47.518    41.407  47.464    41.505  47.411    41.601  47.358   
 41.695  47.306    41.788  47.255    41.879  47.203    41.969  47.153   
 42.057  47.102    42.144  47.052    42.229  47.003    42.313  46.953   
 42.396  46.905    42.477  46.856    42.557  46.808    42.636  46.761   
 42.713  46.713    42.789  46.667    42.864  46.620    42.937  46.574   
 43.010  46.528    43.081  46.483    43.152  46.438    43.221  46.393   
 43.289  46.349    43.356  46.305    43.422  46.261    43.487  46.218   
 43.551  46.175    43.615  46.132    43.677  46.090    43.738  46.048   
 43.799  46.007    43.858  45.965    43.917  45.924    43.975  45.883   
 44.031  45.843    44.088  45.803    44.143  45.763    44.198  45.723   
 44.251  45.684    44.304  45.645    44.357  45.606    44.408  45.568   
 44.459  45.530    44.514  45.489    44.575  45.442    44.636  45.395   
 44.696  45.349    44.754  45.303    44.812  45.258    44.868  45.213   
 44.924  45.169    44.979  45.125    45.033  45.081    45.086  45.038   
 45.138  44.995    45.189  44.952    45.240  44.910    45.290  44.868   
 45.339  44.826    45.387  44.785    45.434  44.744    45.481  44.704   
 45.527  44.664    45.572  44.624    45.617  44.584    45.660  44.545   
 45.704  44.506    45.746  44.468    45.788  44.429    45.830  44.391   
 45.870  44.354    45.910  44.316    45.950  44.279    45.989  44.243   
 46.027  44.206    46.065  44.170    46.102  44.134    46.139  44.099   
 46.175  44.063    46.211  44.028    46.254  43.986    46.297  43.943   
 46.340  43.900    46.381  43.858    46.422  43.816    46.463  43.774   
 46.502  43.733    46.541  43.693    46.579  43.653    46.617  43.613   
 46.654  43.573    46.690  43.534    46.726  43.495    46.761  43.457   
 46.796  43.418    46.830  43.381   ] scurve 
[  46.830  43.381    46.864  43.343    46.897  43.306    46.929  43.269   
 46.961  43.233    46.993  43.197    47.024  43.161    47.054  43.125   
 47.084  43.090    47.114  43.055    47.143  43.021    47.172  42.986   
 47.200  42.952    47.228  42.919    47.255  42.885    47.282  42.852   
 47.309  42.819    47.335  42.787    47.361  42.754    47.386  42.722   
 47.411  42.690    47.439  42.655    47.470  42.616    47.500  42.577   
 47.529  42.539    47.558  42.501    47.586  42.463    47.614  42.426   
 47.641  42.390    47.668  42.353    47.694  42.317    47.720  42.282   
 47.746  42.246    47.771  42.211    47.795  42.177    47.820  42.142   
 47.843  42.108    47.867  42.075    47.890  42.041    47.913  42.008   
 47.935  41.976    47.957  41.943    47.978  41.911    48.000  41.879   
 48.021  41.848    48.041  41.816    48.062  41.785    48.081  41.755   
 48.101  41.724    48.120  41.694    48.140  41.664    48.158  41.635   
 48.177  41.605    48.195  41.576    48.213  41.548    48.231  41.519   
 48.248  41.491    48.266  41.462    48.287  41.427    48.307  41.393   
 48.328  41.359    48.348  41.325    48.367  41.292    48.386  41.259   
 48.405  41.226    48.424  41.194    48.442  41.162    48.460  41.131   
 48.478  41.099    48.495  41.068    48.512  41.038    48.529  41.007   
 48.545  40.977    48.561  40.948    48.577  40.918    48.593  40.889   
 48.608  40.860    48.624  40.831    48.638  40.803    48.653  40.775   
 48.667  40.747    48.682  40.720    48.696  40.693    48.709  40.666   
 48.723  40.639    48.736  40.612    48.749  40.586    48.762  40.560   
 48.775  40.534    48.787  40.509    48.800  40.484    48.812  40.459   
 48.824  40.434    48.836  40.409    48.849  40.381    48.863  40.351   
 48.877  40.321    48.891  40.292    48.904  40.262    48.917  40.234   
 48.930  40.205    48.943  40.177    48.956  40.149    48.968  40.122   
 48.980  40.094    48.992  40.067    49.003  40.041    49.015  40.014   
 49.026  39.988    49.037  39.962    49.048  39.936    49.059  39.911   
 49.069  39.886    49.080  39.861    49.090  39.837    49.100  39.812   
 49.110  39.788    49.119  39.764    49.129  39.741    49.138  39.717   
 49.147  39.694    49.156  39.671    49.165  39.648    49.174  39.626   
 49.183  39.604    49.191  39.582    49.200  39.560    49.208  39.538   
 49.216  39.517    49.224  39.496    49.232  39.473    49.242  39.447   
 49.251  39.421    49.261  39.396    49.270  39.371    49.279  39.346   
 49.288  39.321    49.296  39.297    49.305  39.273    49.313  39.249   
 49.321  39.226    49.329  39.203    49.337  39.180    49.345  39.157   
 49.352  39.135    49.360  39.112    49.367  39.090    49.375  39.069   
 49.382  39.047    49.389  39.026    49.395  39.005    49.402  38.984   
 49.409  38.963    49.415  38.943    49.422  38.923    49.428  38.903   
 49.434  38.883    49.440  38.863    49.446  38.844    49.452  38.825   
 49.458  38.805    49.464  38.787    49.470  38.768    49.475  38.750   
 49.481  38.731    49.486  38.713    49.491  38.695    49.497  38.674   
 49.504  38.652    49.510  38.630    49.516  38.609    49.522  38.588   
 49.528  38.567    49.534  38.546    49.540  38.526    49.545  38.505   
 49.551  38.485    49.556  38.466    49.562  38.446    49.567  38.427   
 49.572  38.408    49.577  38.389    49.582  38.370    49.587  38.352   
 49.591  38.334    49.596  38.316    49.601  38.298    49.605  38.280   
 49.610  38.263    49.614  38.245   ] scurve 
[  49.614  38.245    49.618  38.228    49.623  38.211    49.627  38.194   
 49.631  38.178    49.635  38.161    49.639  38.145    49.643  38.129   
 49.647  38.113    49.650  38.097    49.654  38.082    49.658  38.066   
 49.661  38.051    49.665  38.036    49.669  38.019    49.673  38.001   
 49.677  37.982    49.681  37.964    49.685  37.947    49.689  37.929   
 49.693  37.912    49.697  37.894    49.701  37.877    49.704  37.860   
 49.708  37.844    49.711  37.827    49.715  37.811    49.718  37.795   
 49.722  37.779    49.725  37.764    49.728  37.748    49.731  37.733   
 49.734  37.718    49.738  37.703    49.741  37.688    49.744  37.673   
 49.746  37.659    49.749  37.644    49.752  37.630    49.755  37.616   
 49.758  37.602    49.760  37.588    49.763  37.575    49.765  37.561   
 49.768  37.548    49.771  37.535    49.773  37.522    49.775  37.509   
 49.778  37.496    49.780  37.483    49.782  37.471    50.000  35.000
] scurve 
  6.440  55.710 moveto 
[   6.440  55.710     5.098  55.863     4.011  55.982   ] scurve 
  6.440  41.430 moveto 
[   6.440  41.430     7.723  41.408     8.553  41.393     9.354  41.378   
 10.127  41.363    10.875  41.349    11.598  41.334    12.298  41.319   
 12.975  41.305    13.631  41.290    14.266  41.276    14.882  41.262   
 15.480  41.248    16.059  41.233    16.622  41.219    17.169  41.205   
 17.699  41.191    18.215  41.177    18.717  41.164    19.205  41.150   
 19.680  41.136    20.142  41.122    20.591  41.109    21.029  41.095   
 21.456  41.082    21.872  41.069    22.277  41.055    22.673  41.042   
 23.059  41.029    23.435  41.015    23.802  41.002    24.161  40.989   
 24.511  40.976    24.853  40.963    25.187  40.950    25.513  40.938   
 25.833  40.925    26.145  40.912    26.450  40.900    26.748  40.887   
 27.040  40.874    27.326  40.862    27.606  40.849    27.880  40.837   
 28.148  40.825    28.411  40.812    28.668  40.800    28.920  40.788   
 29.167  40.776    29.409  40.764    29.647  40.752    29.879  40.740   
 30.108  40.728    30.332  40.716    30.551  40.704    30.767  40.692   
 30.978  40.681    31.186  40.669    31.390  40.657    31.590  40.646   
 31.786  40.634    31.979  40.623    32.169  40.611    32.355  40.600   
 32.538  40.588    32.717  40.577    32.894  40.566    33.068  40.555   
 33.238  40.543    33.406  40.532    33.571  40.521    33.733  40.510   
 33.893  40.499    34.050  40.488    34.204  40.477    34.356  40.467   
 34.506  40.456    34.653  40.445    34.798  40.434    34.940  40.423   
 35.081  40.413    35.219  40.402    35.355  40.392    35.489  40.381   
 35.621  40.371    35.751  40.360    35.879  40.350    36.005  40.339   
 36.129  40.329    36.251  40.319    36.372  40.309    36.491  40.298   
 36.608  40.288    36.724  40.278    36.838  40.268    36.950  40.258   
 37.061  40.248    37.170  40.238    37.277  40.228    37.384  40.218   
 37.488  40.208    37.592  40.199    37.693  40.189    37.794  40.179   
 37.893  40.169    37.991  40.160    38.088  40.150    38.183  40.140   
 38.277  40.131    38.370  40.121    38.461  40.112    38.552  40.102   
 38.641  40.093    38.729  40.084    38.816  40.074    38.902  40.065   
 38.987  40.056    39.071  40.046    39.154  40.037    39.235  40.028   
 39.316  40.019    39.396  40.010    39.475  40.001    39.552  39.991   
 39.629  39.982    39.705  39.973    39.780  39.965    39.865  39.954   
 39.956  39.943    40.046  39.932    40.135  39.921    40.222  39.910   
 40.309  39.899    40.394  39.889    40.477  39.878    40.560  39.867   
 40.641  39.856    40.722  39.846    40.801  39.835    40.879  39.824   
 40.956  39.814    41.032  39.803    41.107  39.793    41.181  39.783   
 41.254  39.772    41.326  39.762    41.397  39.752    41.467  39.741   
 41.536  39.731    41.605  39.721    41.672  39.711    41.739  39.701   
 41.804  39.691    41.869  39.681    41.933  39.671    41.996  39.661   
 42.059  39.651    42.120  39.642    42.181  39.632    42.242  39.622   
 42.301  39.613    42.360  39.603    42.418  39.593    42.476  39.584   
 42.547  39.572    42.616  39.560    42.685  39.548    42.752  39.536   
 42.819  39.525    42.884  39.513    42.949  39.502    43.013  39.490   
 43.075  39.479    43.137  39.468    43.198  39.456    43.258  39.445   
 43.317  39.434    43.376  39.423    43.434  39.412    43.490  39.401   
 43.546  39.390    43.602  39.379    43.656  39.368    43.710  39.357   
 43.763  39.347    43.816  39.336    43.867  39.325    43.918  39.315   
 43.969  39.304    44.018  39.294   ] scurve 
[  44.018  39.294    44.067  39.283    44.116  39.273    44.164  39.263   
 44.211  39.252    44.257  39.242    44.303  39.232    44.349  39.222   
 44.394  39.212    44.438  39.202    44.482  39.192    44.532  39.180   
 44.585  39.168    44.637  39.156    44.689  39.143    44.739  39.131   
 44.789  39.119    44.839  39.107    44.887  39.095    44.935  39.083   
 44.982  39.072    45.028  39.060    45.074  39.048    45.119  39.037   
 45.164  39.025    45.207  39.014    45.251  39.002    45.293  38.991   
 45.335  38.980    45.377  38.969    45.417  38.958    45.458  38.947   
 45.497  38.936    45.537  38.925    45.575  38.914    45.613  38.903   
 45.651  38.892    45.688  38.881    45.725  38.871    45.761  38.860   
 45.796  38.850    45.832  38.839    45.866  38.829    45.901  38.818   
 45.935  38.808    45.968  38.798    46.001  38.788    46.035  38.777   
 46.075  38.765    46.115  38.752    46.153  38.740    46.192  38.727   
 46.229  38.715    46.266  38.703    46.303  38.691    46.339  38.679   
 46.374  38.667    46.409  38.655    46.444  38.643    46.478  38.631   
 46.511  38.620    46.544  38.608    46.576  38.597    46.608  38.585   
 46.640  38.574    46.671  38.563    46.701  38.552    46.731  38.540   
 46.761  38.529    46.790  38.518    46.819  38.507    46.848  38.497   
 46.876  38.486    46.904  38.475    46.931  38.464    46.958  38.454   
 46.985  38.443    47.011  38.433    47.037  38.422    47.062  38.412   
 47.087  38.402    47.112  38.392    47.137  38.381    47.161  38.371   
 47.189  38.360    47.219  38.347    47.248  38.335    47.276  38.322   
 47.304  38.310    47.332  38.298    47.359  38.286    47.386  38.274   
 47.412  38.262    47.438  38.251    47.463  38.239    47.488  38.228   
 47.513  38.216    47.537  38.205    47.561  38.193    47.585  38.182   
 47.608  38.171    47.631  38.160    47.654  38.149    47.676  38.138   
 47.698  38.127    47.720  38.116    47.741  38.106    47.762  38.095   
 47.783  38.084    47.803  38.074    47.823  38.063    47.843  38.053   
 47.863  38.043    47.882  38.033    47.901  38.022    47.920  38.012   
 47.938  38.002    47.957  37.992    47.975  37.983    47.992  37.973   
 48.011  37.962    48.033  37.950    48.054  37.938    48.075  37.926   
 48.095  37.915    48.116  37.903    48.135  37.891    48.155  37.880   
 48.174  37.868    48.193  37.857    48.212  37.846    48.230  37.835   
 48.248  37.824    48.266  37.813    48.284  37.802    48.301  37.791   
 48.318  37.780    48.334  37.769    48.351  37.759    48.367  37.748   
 48.383  37.738    48.399  37.728    48.414  37.717    48.430  37.707   
 48.445  37.697    48.460  37.687    48.474  37.677    48.489  37.667   
 48.503  37.657    48.517  37.648    48.531  37.638    48.544  37.629   
 48.558  37.619    48.571  37.610    48.584  37.600    48.597  37.591   
 48.609  37.582    48.625  37.571    48.640  37.559    48.655  37.548   
 48.670  37.537    48.684  37.526    48.699  37.515    48.713  37.504   
 48.727  37.493    48.740  37.482    48.754  37.472    48.767  37.461   
 48.780  37.451    48.793  37.440    48.806  37.430    48.818  37.420   
 48.830  37.410    48.842  37.400    48.854  37.390    48.866  37.380   
 48.877  37.370    48.888  37.361    48.900  37.351    48.910  37.342   
 48.921  37.332    48.932  37.323    48.942  37.314    48.953  37.304   
 48.963  37.295    48.973  37.286    48.983  37.277    48.992  37.268   
 49.002  37.259    49.011  37.251   ] scurve 
[  49.011  37.251    49.021  37.242    49.030  37.233    49.039  37.225   
 49.049  37.215    49.060  37.205    49.071  37.194    49.081  37.184   
 49.092  37.173    49.102  37.163    49.112  37.153    49.122  37.143   
 49.132  37.133    49.141  37.124    49.151  37.114    49.160  37.104   
 49.169  37.095    49.178  37.085    49.187  37.076    49.195  37.067   
 49.204  37.057    49.212  37.048    49.220  37.039    49.228  37.030   
 49.236  37.022    49.244  37.013    49.252  37.004    49.260  36.995   
 49.267  36.987    49.275  36.978    49.282  36.970    49.289  36.962   
 49.296  36.953    49.303  36.945    49.310  36.937    49.317  36.929   
 49.323  36.921    49.330  36.913    49.336  36.905    49.343  36.897   
 49.349  36.889    49.357  36.880    50.000  35.000   ] scurve 
  6.440  41.430 moveto 
[   6.440  41.430     5.088  41.452     4.129  41.467   ] scurve 
 43.570  55.710 moveto 
[  43.570  55.710    44.045  55.030    44.255  54.719    44.454  54.418   
 44.643  54.125    44.822  53.841    44.992  53.566    45.155  53.298   
 45.309  53.039    45.456  52.786    45.596  52.540    45.730  52.301   
 45.857  52.069    45.979  51.842    46.096  51.622    46.208  51.407   
 46.315  51.198    46.417  50.994    46.516  50.795    46.610  50.601   
 46.700  50.412    46.787  50.227    46.871  50.046    46.951  49.870   
 47.028  49.698    47.102  49.530    47.174  49.366    47.243  49.205   
 47.309  49.048    47.373  48.894    47.435  48.744    47.495  48.597   
 47.552  48.453    47.608  48.312    47.662  48.174    47.713  48.038   
 47.764  47.906    47.812  47.776    47.859  47.649    47.905  47.524   
 47.949  47.402    47.991  47.282    48.033  47.164    48.073  47.049   
 48.112  46.936    48.149  46.825    48.186  46.715    48.222  46.608   
 48.256  46.503    48.290  46.400    48.322  46.299    48.354  46.199   
 48.384  46.101    48.414  46.005    48.443  45.910    48.472  45.818   
 48.499  45.726    48.526  45.636    48.552  45.548    48.577  45.461   
 48.602  45.376    48.626  45.292    48.649  45.209    48.672  45.128   
 48.695  45.047    48.716  44.969    48.737  44.891    48.758  44.815   
 48.778  44.739    48.798  44.665    48.817  44.592    48.836  44.520   
 48.854  44.450    48.872  44.380    48.889  44.311    48.906  44.243   
 48.923  44.176    48.939  44.111    48.955  44.046    48.971  43.982   
 48.986  43.919    49.001  43.857    49.016  43.795    49.030  43.735   
 49.044  43.675    49.057  43.616    49.071  43.558    49.084  43.501   
 49.097  43.444    49.109  43.389    49.121  43.334    49.133  43.279   
 49.145  43.226    49.157  43.173    49.168  43.121    49.179  43.069   
 49.190  43.018    49.200  42.968    49.211  42.918    49.221  42.869   
 49.231  42.821    49.241  42.773    49.251  42.726    49.260  42.679   
 49.269  42.633    49.278  42.587    49.287  42.542    49.296  42.498   
 49.305  42.454    49.313  42.410    49.322  42.367    49.330  42.325   
 49.338  42.283    49.346  42.241    49.353  42.200    49.361  42.160   
 49.368  42.119    49.376  42.080    49.383  42.041    49.390  42.002   
 49.397  41.963    49.404  41.925    49.411  41.888    49.417  41.851   
 49.424  41.814    49.430  41.778    49.436  41.742    49.442  41.706   
 49.449  41.671    49.455  41.636    49.460  41.601    49.466  41.567   
 49.472  41.534    49.478  41.500    49.483  41.467    49.490  41.428   
 49.496  41.387    49.503  41.347    49.509  41.308    49.515  41.269   
 49.522  41.231    49.528  41.193    49.533  41.155    49.539  41.118   
 49.545  41.082    49.551  41.045    49.556  41.010    49.561  40.974   
 49.567  40.939    49.572  40.905    49.577  40.871    49.582  40.837   
 49.587  40.804    49.592  40.771    49.597  40.738    49.601  40.706   
 49.606  40.674    49.610  40.643    49.615  40.611    49.619  40.581   
 49.623  40.550    49.628  40.520    49.632  40.490    49.636  40.461   
 49.640  40.431    49.644  40.403    49.648  40.374    49.652  40.346   
 49.655  40.318    49.659  40.290    49.663  40.263    49.667  40.233   
 49.671  40.200    49.675  40.167    49.679  40.134    49.684  40.102   
 49.688  40.070    49.692  40.039    49.696  40.008    49.699  39.977   
 49.703  39.947    49.707  39.917    49.710  39.887    49.714  39.858   
 49.717  39.829    49.721  39.801    49.724  39.772    49.728  39.745   
 49.731  39.717    49.734  39.690   ] scurve 
[  49.734  39.690    49.737  39.663    49.740  39.636    49.743  39.610   
 49.746  39.584    49.749  39.558    49.752  39.533    49.755  39.508   
 49.758  39.483    49.760  39.458    49.763  39.434    49.766  39.410   
 49.768  39.386    49.771  39.363    49.773  39.339    49.776  39.316   
 49.778  39.294    49.780  39.271    49.783  39.249    49.786  39.222   
 49.788  39.194    49.791  39.168    49.794  39.141    49.797  39.115   
 49.799  39.089    49.802  39.063    49.804  39.038    49.807  39.013   
 49.809  38.989    49.812  38.964    49.814  38.940    49.816  38.917   
 49.818  38.893    49.821  38.870    49.823  38.847    49.825  38.824   
 49.827  38.802    49.829  38.780    49.831  38.758    50.000  35.000
] scurve 
 43.570  55.710 moveto 
[  43.570  55.710    43.051  56.414    42.828  56.706    42.594  57.007   
 42.349  57.316    42.091  57.634    41.819  57.961    41.533  58.297   
 41.232  58.644    40.914  59.000    40.579  59.368    40.224  59.748   
 39.870  60.117    39.555  60.439    39.224  60.769    38.878  61.107   
 38.515  61.455    38.133  61.812    37.732  62.179    37.309  62.556   
 36.865  62.943    36.396  63.342    35.901  63.752    35.379  64.174   
 34.929  64.528    34.468  64.884    33.986  65.248    33.481  65.620   
] scurve 
 52.860  60.710 moveto 
[  52.860  60.710    52.641  59.648    52.507  58.978    52.382  58.344   
 52.267  57.743    52.160  57.171    52.061  56.628    51.968  56.111   
 51.881  55.617    51.800  55.147    51.724  54.697    51.653  54.267   
 51.586  53.856    51.523  53.461    51.463  53.083    51.408  52.720   
 51.355  52.371    51.305  52.036    51.258  51.713    51.213  51.403   
 51.171  51.104    51.131  50.815    51.093  50.537    51.057  50.268   
 51.022  50.008    50.989  49.757    50.958  49.515    50.929  49.280   
 50.900  49.053    50.873  48.833    50.847  48.619    50.822  48.412   
 50.799  48.211    50.776  48.017    50.754  47.828    50.733  47.644   
 50.713  47.465    50.694  47.292    50.676  47.123    50.658  46.959   
 50.641  46.799    50.625  46.643    50.609  46.492    50.594  46.344   
 50.579  46.200    50.565  46.060    50.552  45.923    50.539  45.789   
 50.526  45.659    50.514  45.532    50.502  45.408    50.491  45.287   
 50.480  45.168    50.469  45.052    50.459  44.939    50.449  44.829   
 50.439  44.720    50.430  44.614    50.421  44.511    50.412  44.409   
 50.404  44.310    50.396  44.213    50.388  44.118    50.380  44.025   
 50.373  43.933    50.365  43.844    50.358  43.756    50.351  43.670   
 50.345  43.586    50.338  43.503    50.332  43.422    50.326  43.343   
 50.320  43.264    50.314  43.188    50.308  43.113    50.303  43.039   
 50.298  42.966    50.292  42.895    50.287  42.825    50.282  42.756   
 50.278  42.689    50.273  42.622    50.268  42.557    50.264  42.493   
 50.260  42.430    50.255  42.368    50.251  42.307    50.247  42.247   
 50.243  42.188    50.239  42.130    50.236  42.073    50.232  42.017   
 50.228  41.961    50.225  41.907    50.221  41.853    50.218  41.800   
 50.215  41.748    50.212  41.697    50.208  41.647    50.205  41.597   
 50.202  41.548    50.199  41.500    50.197  41.453    50.194  41.406   
 50.191  41.360    50.188  41.314    50.186  41.269    50.183  41.225   
 50.181  41.181    50.178  41.138    50.176  41.096    50.173  41.054   
 50.171  41.013    50.169  40.972    50.167  40.932    50.164  40.893   
 50.162  40.854    50.160  40.815    50.158  40.777    50.156  40.739   
 50.154  40.702    50.152  40.666    50.150  40.630    50.148  40.594   
 50.147  40.559    50.145  40.524    50.143  40.490    50.141  40.456   
 50.140  40.422    50.138  40.389    50.136  40.356    50.135  40.324   
 50.133  40.292    50.131  40.261    50.130  40.230    50.128  40.195   
 50.126  40.157    50.125  40.119    50.123  40.082    50.121  40.046   
 50.119  40.010    50.000  35.000   ] scurve 
 52.860  60.710 moveto 
[  52.860  60.710    53.107  61.864    53.279  62.644    53.439  63.353   
 53.611  64.100    53.797  64.888    53.997  65.719   ] scurve 
 71.420  64.990 moveto 
[  71.420  64.990    70.161  63.615    69.020  62.361    67.981  61.211   
 67.031  60.155    66.161  59.180    65.361  58.279    64.623  57.442   
 63.941  56.663    63.199  55.809    62.480  54.976    61.809  54.191   
 61.063  53.311    60.389  52.508    59.779  51.773    59.223  51.097   
 58.716  50.474    58.251  49.896    57.824  49.361    57.431  48.862   
 57.068  48.397    56.732  47.963    56.420  47.555    56.130  47.172   
 55.860  46.812    55.609  46.473    55.373  46.153    55.153  45.850   
 54.946  45.563    54.752  45.291    54.569  45.032    54.397  44.786   
 54.234  44.552    54.081  44.329    53.936  44.116    53.799  43.913   
 53.669  43.718    53.546  43.532    53.429  43.354    53.318  43.182   
 53.212  43.018    53.112  42.860    53.016  42.709    52.924  42.563   
 52.837  42.422    52.754  42.287    52.674  42.156    52.598  42.030   
 52.525  41.909    52.455  41.791    52.388  41.678    52.324  41.568   
 52.262  41.462    52.203  41.359    52.146  41.259    52.091  41.162   
 52.039  41.069    51.988  40.978    51.940  40.890    51.893  40.804   
 51.848  40.721    51.804  40.640    51.762  40.562    51.721  40.485   
 51.682  40.411    51.644  40.339    51.608  40.268    51.572  40.200   
 51.533  40.122    51.492  40.042    51.453  39.963    51.415  39.887   
 51.379  39.814    51.344  39.742    51.310  39.673    51.278  39.606   
 51.247  39.540    51.217  39.477    51.188  39.415    51.160  39.355   
 51.133  39.296    51.107  39.239    51.082  39.183    51.058  39.129   
 51.034  39.077    51.011  39.025    50.990  38.975    50.968  38.926   
 50.948  38.879    50.928  38.832    50.909  38.787    50.890  38.742   
 50.872  38.699    50.854  38.657    50.837  38.615    50.821  38.575   
 50.805  38.535    50.789  38.497    50.774  38.459    50.760  38.422   
 50.745  38.386    50.731  38.350    50.718  38.315    50.705  38.281   
 50.692  38.248    50.680  38.215    50.668  38.183    50.654  38.145   
 50.639  38.107    50.626  38.070    50.612  38.033    50.600  37.998   
 50.587  37.963    50.575  37.929    50.563  37.896    50.552  37.864   
 50.541  37.832    50.530  37.801    50.520  37.770    50.510  37.741   
 50.500  37.712    50.491  37.683    50.481  37.655    50.472  37.628   
 50.464  37.601    50.455  37.575    50.447  37.549    50.439  37.524   
 50.431  37.500    50.424  37.475    50.416  37.452    50.409  37.428   
 50.402  37.405    50.395  37.383    50.388  37.361    50.382  37.339   
 50.376  37.318    50.369  37.297    50.363  37.277    50.357  37.257   
 50.352  37.237    50.346  37.218    50.341  37.199    50.335  37.180   
 50.329  37.158    50.323  37.136    50.317  37.114    50.311  37.092   
 50.305  37.071    50.299  37.050    50.294  37.030    50.289  37.010   
 50.283  36.990    50.278  36.971    50.273  36.952    50.269  36.934   
 50.264  36.916    50.259  36.898    50.255  36.881    50.251  36.863   
 50.246  36.847    50.242  36.830    50.238  36.814    50.234  36.798   
 50.231  36.782    50.227  36.767    50.223  36.752    50.220  36.737   
 50.216  36.722    50.213  36.708    50.210  36.694    50.206  36.680   
 50.203  36.666    50.200  36.653    50.197  36.639    50.194  36.626   
 50.191  36.614    50.189  36.601    50.186  36.588    50.183  36.576   
 50.180  36.564    50.177  36.550    50.174  36.535    50.171  36.521   
 50.168  36.507    50.165  36.493    50.162  36.480    50.160  36.466   
 50.157  36.453    50.000  35.000   ] scurve 
 30.560   7.170 moveto 
[  30.560   7.170    29.167   5.643    27.600   3.939   ] scurve 
 30.560   7.170 moveto 
[  30.560   7.170    31.807   8.546    32.928   9.793    33.940  10.928   
 34.859  11.964    35.695  12.916    36.459  13.791    37.160  14.600   
 37.860  15.415    38.590  16.272    39.252  17.058    39.855  17.780   
 40.406  18.446    40.911  19.063    41.376  19.635    41.804  20.168   
 42.200  20.665    42.566  21.129    42.906  21.565    43.222  21.974   
 43.517  22.359    43.793  22.722    44.050  23.064    44.292  23.388   
 44.518  23.695    44.731  23.986    44.931  24.262    45.120  24.525   
 45.298  24.775    45.466  25.014    45.625  25.242    45.776  25.459   
 45.919  25.668    46.054  25.867    46.183  26.058    46.306  26.241   
 46.422  26.416    46.533  26.585    46.639  26.748    46.740  26.904   
 46.836  27.054    46.928  27.199    47.016  27.339    47.100  27.473   
 47.181  27.604    47.258  27.729    47.332  27.851    47.403  27.968   
 47.471  28.082    47.537  28.192    47.600  28.299    47.660  28.402   
 47.718  28.502    47.774  28.599    47.830  28.696    47.894  28.810   
 47.956  28.920    48.015  29.026    48.072  29.128    48.126  29.227   
 48.178  29.323    48.227  29.416    48.275  29.505    48.321  29.592   
 48.365  29.676    48.407  29.758    48.447  29.837    48.486  29.914   
 48.524  29.988    48.560  30.060    48.595  30.131    48.629  30.199   
 48.661  30.265    48.692  30.330    48.722  30.393    48.752  30.454   
 48.780  30.514    48.807  30.572    48.833  30.628    48.858  30.683   
 48.883  30.737    48.907  30.790    48.930  30.841    48.952  30.891   
 48.974  30.939    48.995  30.987    49.015  31.034    49.035  31.079   
 49.054  31.123    49.072  31.167    49.090  31.209    49.108  31.251   
 49.125  31.291    49.141  31.331    49.157  31.370    49.177  31.418   
 49.195  31.464    49.214  31.509    49.231  31.553    49.248  31.596   
 49.264  31.638    49.280  31.678    49.295  31.718    49.310  31.757   
 49.325  31.795    49.339  31.832    49.352  31.868    49.365  31.904   
 49.378  31.938    49.390  31.972    49.402  32.005    49.414  32.038   
 49.425  32.069    49.436  32.100    49.447  32.131    49.457  32.161   
 49.467  32.190    49.477  32.218    49.486  32.246    49.495  32.274   
 49.504  32.300    49.513  32.327    49.522  32.353    49.530  32.378   
 49.538  32.403    49.546  32.427    49.554  32.451    49.561  32.475   
 49.568  32.498    49.576  32.520    49.582  32.543    49.589  32.565   
 49.597  32.589    49.605  32.615    49.613  32.641    49.620  32.666   
 49.627  32.691    49.635  32.715    49.641  32.738    49.648  32.761   
 49.655  32.784    49.661  32.806    49.667  32.828    49.673  32.849   
 49.679  32.870    49.685  32.891    49.690  32.911    49.696  32.931   
 49.701  32.950    49.706  32.969    49.711  32.988    49.716  33.006   
 49.721  33.024    49.725  33.042    49.730  33.059    49.734  33.076   
 49.739  33.093    49.743  33.109    49.747  33.125    49.751  33.141   
 49.755  33.157    49.758  33.172    49.762  33.187    49.766  33.202   
 49.769  33.217    49.773  33.231    49.776  33.245    49.780  33.259   
 49.783  33.273    49.787  33.289    49.790  33.306    49.794  33.322   
 49.798  33.338    49.801  33.353    49.805  33.369    49.808  33.384   
 49.811  33.399    49.815  33.413    49.818  33.427    49.821  33.441   
 49.824  33.455    49.827  33.469    49.830  33.482    49.832  33.495   
 49.835  33.508    49.838  33.520    49.840  33.533    49.843  33.545   
 49.845  33.557    50.000  35.000   ] scurve 
 60.000  40.000 moveto 
[  60.000  40.000    60.525  39.056    60.632  38.992    60.668  38.971   
 60.706  38.951    60.745  38.931    60.785  38.912    60.826  38.892   
 60.869  38.873    60.912  38.854    60.958  38.835    61.004  38.817   
 61.052  38.798    61.102  38.780    61.152  38.762    61.205  38.744   
 61.259  38.726    61.314  38.708    61.371  38.691    61.429  38.674   
 61.490  38.656    61.552  38.639    61.616  38.623    61.681  38.606   
 61.749  38.589    61.818  38.573    61.889  38.557    61.963  38.541   
 62.038  38.525    62.116  38.509    62.195  38.493    62.278  38.478   
 62.362  38.463    62.449  38.447    62.538  38.432    62.630  38.417   
 62.725  38.403    62.822  38.388    62.922  38.373    63.025  38.359   
 63.132  38.344    63.241  38.330    63.354  38.316    63.470  38.302   
 63.589  38.288    63.713  38.275    63.840  38.261    63.971  38.247   
 64.096  38.235    64.207  38.224    64.321  38.214    64.437  38.203   
 64.557  38.193    64.680  38.182    64.806  38.172    64.935  38.162   
 65.068  38.152    65.205  38.142    65.345  38.132    65.489  38.122   
 65.637  38.112    65.789  38.102    65.945  38.092    66.106  38.082   
 66.272  38.073    66.442  38.063    66.618  38.053    66.798  38.044   
 66.984  38.034    67.176  38.025    67.374  38.016    67.578  38.006   
 67.788  37.997    68.006  37.988    68.230  37.979    68.462  37.970   
 68.693  37.961    68.890  37.954    69.093  37.947    69.301  37.940   
 69.515  37.933    69.735  37.926    69.962  37.919    70.195  37.912   
 70.435  37.905    70.683  37.898    70.938  37.891    71.200  37.884   
 71.471  37.878    71.751  37.871    72.039  37.864    72.337  37.857   
 72.645  37.851    72.963  37.844    73.292  37.838    73.632  37.831   
 73.985  37.824    74.323  37.818    74.624  37.813    74.933  37.808   
 75.252  37.803    75.581  37.798    75.919  37.793    76.268  37.788   
 76.628  37.782    77.000  37.777    77.384  37.772    77.781  37.767   
 78.191  37.762    78.614  37.757    79.053  37.752    79.507  37.747   
 79.978  37.742    80.465  37.737    80.971  37.733    81.419  37.728   
 81.851  37.724    82.297  37.721    82.756  37.717    83.231  37.713   
 83.721  37.709    84.227  37.705    84.751  37.701    85.292  37.697   
 85.852  37.694    86.432  37.690    87.033  37.686    87.656  37.682   
 88.303  37.678    88.974  37.675    89.658  37.671    90.235  37.668   
 90.831  37.665    91.446  37.662    92.081  37.659    92.738  37.656   
 93.417  37.653    94.120  37.650    94.847  37.647    95.601  37.644   
] scurve 
 60.000  40.000 moveto 
[  60.000  40.000    60.534  41.476    60.689  41.721    60.841  41.948   
 61.011  42.191    61.202  42.452    61.415  42.732    61.652  43.034   
 61.916  43.360    62.210  43.714    62.537  44.100    62.902  44.521   
 63.309  44.983    63.765  45.492    64.278  46.056    64.898  46.730   
 65.739  47.630    66.731  48.680    67.916  49.921    69.351  51.409   
 71.117  53.227    73.338  55.498    76.207  58.415    80.045  62.300   
 85.432  67.731   ] scurve 
 40.000  30.000 moveto 
[  40.000  30.000    39.466  28.524    39.311  28.279    39.159  28.052   
 38.989  27.809    38.798  27.548    38.585  27.268    38.348  26.966   
 38.084  26.640    37.790  26.286    37.463  25.900    37.098  25.479   
 36.691  25.017    36.235  24.508    35.722  23.944    35.102  23.270   
 34.261  22.370    33.269  21.320    32.084  20.079    30.649  18.591   
 28.883  16.773    26.662  14.502    23.793  11.585    19.955   7.700   
 14.568   2.269   ] scurve 
 40.000  30.000 moveto 
[  40.000  30.000    39.475  30.944    39.368  31.008    39.332  31.029   
 39.294  31.049    39.255  31.069    39.215  31.088    39.174  31.108   
 39.131  31.127    39.088  31.146    39.042  31.165    38.996  31.183   
 38.948  31.202    38.898  31.220    38.848  31.238    38.795  31.256   
 38.741  31.274    38.686  31.292    38.629  31.309    38.571  31.326   
 38.510  31.344    38.448  31.361    38.384  31.377    38.319  31.394   
 38.251  31.411    38.182  31.427    38.111  31.443    38.037  31.459   
 37.962  31.475    37.884  31.491    37.805  31.507    37.722  31.522   
 37.638  31.537    37.551  31.553    37.462  31.568    37.370  31.583   
 37.275  31.597    37.178  31.612    37.078  31.627    36.975  31.641   
 36.868  31.656    36.759  31.670    36.646  31.684    36.530  31.698   
 36.411  31.712    36.287  31.725    36.160  31.739    36.029  31.753   
 35.904  31.765    35.793  31.776    35.679  31.786    35.563  31.797   
 35.443  31.807    35.320  31.818    35.194  31.828    35.065  31.838   
 34.932  31.848    34.795  31.858    34.655  31.868    34.511  31.878   
 34.363  31.888    34.211  31.898    34.055  31.908    33.894  31.918   
 33.728  31.927    33.558  31.937    33.382  31.947    33.202  31.956   
 33.016  31.966    32.824  31.975    32.626  31.984    32.422  31.994   
 32.212  32.003    31.994  32.012    31.770  32.021    31.538  32.030   
 31.307  32.039    31.110  32.046    30.907  32.053    30.699  32.060   
 30.485  32.067    30.265  32.074    30.038  32.081    29.805  32.088   
 29.565  32.095    29.317  32.102    29.062  32.109    28.800  32.116   
 28.529  32.122    28.249  32.129    27.961  32.136    27.663  32.143   
 27.355  32.149    27.037  32.156    26.708  32.162    26.368  32.169   
 26.015  32.176    25.677  32.182    25.376  32.187    25.067  32.192   
 24.748  32.197    24.419  32.202    24.081  32.207    23.732  32.212   
 23.372  32.218    23.000  32.223    22.616  32.228    22.219  32.233   
 21.809  32.238    21.386  32.243    20.947  32.248    20.493  32.253   
 20.022  32.258    19.535  32.263    19.029  32.267    18.581  32.272   
 18.149  32.276    17.703  32.279    17.244  32.283    16.769  32.287   
 16.279  32.291    15.773  32.295    15.249  32.299    14.708  32.303   
 14.148  32.306    13.568  32.310    12.967  32.314    12.344  32.318   
 11.697  32.322    11.026  32.325    10.342  32.329     9.765  32.332   
  9.169  32.335     8.554  32.338     7.919  32.341     7.262  32.344   
  6.583  32.347     5.880  32.350     5.153  32.353     4.399  32.356   
] scurve 
grestore}
\LabelTeX   98.000  31.200 $x$\ELTX
\LabelTeX   46.500  68.000 $y$\ELTX
\EndFig
\vskip8mm

\state Autre M\'ethode de r\'esolution -- |  Utilisation des coordonn\'ees
polaires.

\noindent Pour $r>0$ et $\theta\in\bR$ on pose
$$
\cases{
x=r\,\cos\,\theta\hfill\cr
\noalign{\vskip3pt}
y=r\,\sin\,\theta\hfill\cr
}.
$$
Il vient
$$
\frac{dy}{dx}=\frac{dr\,\sin\,\theta+r\,\cos\,\theta\,d\theta}{dr\,\cos\,\theta-r
\,\sin\,\theta\,d\theta}=\frac{dr\,\tan\,\theta+r\,d\theta}{dr-r\,\tan\,\theta\,d\theta}.
$$
L'\'equation (E) $y'=f\Big(\dfrac{y}{x}\Big)$ se transforme alors en
$$
\eqalign{
&dr\,\tan\,\theta+r\,d\theta=(dr-r\,\tan\,\theta\, d\theta)f(\tan\,\theta),\cr
&dr(f(\tan\,\theta)-\tan\,\theta)=rd\theta(1+\tan\,\theta\,f(\tan\,\theta)),\cr
&\frac{dr}{r}=\frac{1+\tan\,\theta\,f(\tan\,\theta)}{f(\tan\,\theta)-\tan\,\theta}\
d\theta.\cr 
}
$$
On aboutit donc \`a une \'equation \`a variables s\'epar\'ees $r,\theta$. Les
int\'egrales singuli\`eres correspondent aux droites $\theta=\theta_j$ telles que
$f(\tan\,\theta_j)=\tan\,\theta_j$.

\bigskip
\majedeb
\state Exercice -- |  {\it R\'esoudre $y'=\dfrac{x+y}{x-y}$ \`a l'aide des deux
m\'ethodes propos\'ees. Quelle est la nature des courbes int\'egrales~?} 
\par\majefin\par

\section 2. \'Equations du premier ordre non r\'esolues en $y'$|

\subsection 2.1. D\'efinitions et premi\`eres propri\'et\'es|

\noindent On appelle \'equation du premier ordre non r\'esolue en $y'$ une \'equation de
la forme
$$
f(x,y,y')=0\leqno{\rm (E)}
$$
o\`u $(x,y,p)\mapsto f(x,y,p)$ est une fonction de classe $C^1$
dans un ouvert \hbox{$U\subset \bR^3$}. Pla\-\c{c}ons-nous au voisinage d'un
point $(x_0,y_0)\in\bR^2$. On suppose que l'\'equation \hbox{$f(x_0,y_0,p)=0$} 
admet des racines $p_1,p_2,\dots,p_N$ et que ces racines sont simples,
c'est-\`a-dire
$$
\frac{\partial f}{\partial p}\ (x_0,y_0,p_j)\not=0.
$$
D'apr\`es le th\'eor\`eme des fonctions implicites, on sait alors
qu'il existe un voisinage $V$ de $(x_0,y_0)$, un r\'eel $h>0$ et une
fonction $g_j:V\rightarrow\ ]p_j-h,p_j+h[$ de classe $C^1$, $1\leq
j\leq N$, tels que pour tout $(x,y,p)\in V\times\ ]p_j-h,p_j+h[$ on
ait
$$
f(x,y,p)=0\quad\Leftrightarrow\quad p=g_j(x,y).
$$
L'\'equation diff\'erentielle $f(x,y,y')=0$ nous am\`ene alors \`a r\'esoudre
dans $V$ les $N$ \'equations diff\'erentielles 
$$
y'=g_j(x,y).\leqno({\rm E}_j)
$$
Comme $g_j$ est de classe $C^1$, on voit que par tout point $(x,y)\in V$
il passe exactement $N$ courbes int\'egrales dont les pentes sont
les racines $p$ de $f(x,y,p)=0$.

\InsertFig 24.000 47.000 {
1 mm unit
  0.000   5.000 moveto 
 80.000   0.000 vector 
 10.000   0.000 moveto 
 35.000  90.000 vector 
 40.000  20.000 moveto 
  0.700 disk 
 10.000  20.000 moveto 
  0.340 setlinewidth 
  0.700   0.000 indent 
 40.000   5.000 moveto 
  0.700  90.000 indent 
 20.000  12.000 moveto 
[  20.000  12.000   30.000  14.500   40.000  20.000   50.000  30.000  
 60.000  35.000  ] curve 
stroke 
 20.000  27.500 moveto 
[  20.000  27.500   30.000  29.000   40.000  20.000    50.000   4.500
 60.000   2.000  ] curve 
stroke 
 40.000  20.000 moveto 
  0.113 setlinewidth 
 13.000  40.365 vector 
 13.000 220.365 vector 
 13.000 126.529 vector 
 13.000 -53.471 vector 
 40.000   6.500 moveto 
[   2.000   1.000 ] 0 setdash 
 40.000  19.500 lineto 
stroke 
 11.500  20.000 moveto 
 39.500  20.000 lineto 
stroke}
\LabelTeX    6.500  20.000 $y$\ELTX
\LabelTeX   39.200   1.300 $x$\ELTX
\LabelTeX   49.000  12.000 $\hbox{\rm pente}~p_1$\ELTX
\LabelTeX   51.700  26.500 $\hbox{\rm pente}~p_2$\ELTX
\EndFig
\vskip4mm

\state Remarque -- |  Dans cette situation, il arrive fr\'equemment
qu'on ait une famille de courbes int\'egrales $C_{\lambda}$ admettant
une {\it enveloppe} $\Gamma$, c'est-\`a-dire une courbe $\Gamma$ qui
est tangente en chacun de ses points \`a l'une des courbes
$C_{\lambda}$.

\vbox{
\InsertFig -24.000 103.000 {
1 mm unit
gsave 
 65.000  50.000 moveto 
 65.000  45.000 rect  clip  newpath 
110.636  49.909 moveto 
[ 110.636  49.909   109.306  50.561   107.996  51.234   106.708  51.928   
105.442  52.643   104.200  53.382   102.984  54.146   101.795  54.936   
100.635  55.754    99.506  56.601    98.410  57.481    97.349  58.395   
 96.327  59.346    95.346  60.337    94.410  61.371    93.524  62.453   
 92.691  63.587    91.919  64.778    91.213  66.034    90.582  67.361   
 90.036  68.769    89.588  70.269    89.252  71.877    89.049  73.611   
 89.006  75.497    89.149  77.468    89.437  79.270    89.836  80.948   
 90.320  82.530    90.874  84.034    91.488  85.473    92.151  86.856   
 92.857  88.193    93.602  89.488    94.380  90.746    95.189  91.971   
 96.024  93.167    96.885  94.336    97.768  95.481   ] scurve 
122.705  95.205 moveto 
[ 122.705  95.205   121.712  94.214   120.716  93.226   119.717  92.242   
118.714  91.261   117.708  90.285   116.699  89.312   115.686  88.345   
114.668  87.382   113.647  86.423   112.621  85.471   111.590  84.524   
110.555  83.583   109.514  82.649   108.467  81.722   107.413  80.802   
106.353  79.891   105.286  78.989   104.210  78.097   103.126  77.216   
102.032  76.347   100.927  75.492    99.811  74.652    98.680  73.830   
 97.534  73.029    96.370  72.251    95.185  71.502    93.975  70.788   
 92.734  70.118    91.453  69.506    90.121  68.972    88.716  68.556   
 87.198  68.339    85.473  68.554    84.163  69.339    83.296  70.551   
 82.775  72.097    82.532  73.914    82.521  75.956    82.685  77.994   
 82.943  79.875    83.262  81.663    83.623  83.389    84.018  85.068   
 84.438  86.712    84.880  88.327    85.340  89.918    85.815  91.488   
 86.304  93.041    86.805  94.579    87.316  96.104   ] scurve 
 95.168  95.278 moveto 
[  95.168  95.278    94.474  93.995    93.778  92.717    93.078  91.444   
 92.375  90.176    91.667  88.913    90.955  87.657    90.238  86.408   
 89.516  85.167    88.788  83.934    88.053  82.711    87.311  81.500   
 86.560  80.301    85.800  79.118    85.028  77.954    84.243  76.811   
 83.442  75.697    82.621  74.619    81.773  73.592    80.889  72.639   
 79.951  71.808    78.924  71.230    77.707  71.520    77.084  73.382   
 77.006  75.524    77.137  77.696    77.351  79.660    77.610  81.529   
 77.900  83.340    78.212  85.111    78.540  86.854    78.883  88.573   
 79.237  90.274    79.600  91.959    79.972  93.632    80.352  95.293   
 80.738  96.945   ] scurve 
 85.519  95.186 moveto 
[  85.519  95.186    84.955  93.778    84.388  92.375    83.817  90.978   
 83.244  89.586    82.666  88.201    82.084  86.823    81.496  85.453   
 80.904  84.092    80.305  82.743    79.699  81.407    79.084  80.086   
 78.460  78.785    77.823  77.509    77.171  76.265    76.499  75.066   
 75.800  73.937    75.060  72.926    74.250  72.179    73.301  72.553   
 73.000  74.957    73.097  77.296    73.283  79.324    73.509  81.243   
 73.761  83.102    74.030  84.922    74.314  86.715    74.608  88.486   
 74.911  90.240    75.222  91.981    75.539  93.710    75.862  95.429   
] scurve 
 78.274  94.689 moveto 
[  78.274  94.689    77.799  93.196    77.322  91.708    76.842  90.225   
 76.358  88.749    75.871  87.280    75.379  85.819    74.883  84.368   
 74.381  82.928    73.872  81.502    73.355  80.094    72.830  78.708   
 72.292  77.351    71.739  76.036    71.164  74.786    70.558  73.650   
 69.899  72.779    69.146  73.196    69.018  75.919    69.154  78.132   
 69.342  80.136    69.557  82.056    69.789  83.929    70.033  85.770   
 70.288  87.588    70.551  89.389    70.820  91.176    71.095  92.951   
 71.374  94.717    71.658  96.475   ] scurve 
  0.340 setlinewidth
102.371  49.953 moveto 
[ 102.371  49.953   101.653  51.201   100.918  52.432   100.165  53.644   
 99.394  54.837    98.603  56.008    97.791  57.157    96.956  58.282   
 96.096  59.381    95.209  60.453    94.294  61.494    93.347  62.504   
 92.366  63.480    91.348  64.419    90.290  65.318    89.186  66.175   
 88.034  66.986    86.828  67.749    85.563  68.461    84.233  69.119   
 82.834  69.722    81.362  70.269    79.813  70.759    78.188  71.194   
 76.491  71.576    74.727  71.910    72.905  72.199    71.035  72.449   
 69.125  72.664    67.185  72.851    65.220  73.013    63.237  73.155   
 61.240  73.279    59.232  73.388    57.216  73.485    55.193  73.572   
 53.166  73.649    51.135  73.719    49.102  73.782   ] scurve 
grestore}
\LabelTeX   61.000  72.500 $\Gamma$\ELTX
\LabelTeX  107.102  75.782 $C_\lambda$\ELTX
\EndFig
\vskip-43mm
}

\noindent La courbe $\Gamma$ est alors elle-m\^eme une courbe
int\'egrale, car en chaque point sa tangente appartient au champ des
tangentes de l'\'equation (E) (elle co\"{\i}ncide avec la tangente de
l'une des courbes $C_{\lambda}$). $\Gamma$ est donc une solution
singuli\`ere. On notera qu'une telle courbe $\Gamma$ doit satisfaire
simultan\'ement les deux \'equations $f(x,y,y')=0$ et $\partial
f/\partial p(x,y,y')=0$~: chaque point $(x,y)\in\Gamma$ est en effet
limite d'une suite de points en lesquels deux tangentes du champ
viennent se confondre, de sorte que $p=y'$ est racine double de
$f(x,y,p)=0$. En particulier les hypoth\`eses faites ci-dessus pour
appliquer le th\'eor\`eme des fonctions implicites ne sont pas
satisfaites si $(x_0,y_0)\in\Gamma$.

\medskip
\state M\'ethode de R\'esolution -- |  Pour r\'esoudre les \'equations
diff\'erentielles non r\'esolues en $y'$, le principe g\'en\'eral est
de chercher une param\'etrisation de $x,y,y'$ en fonction d'un
param\`etre $t$ qui sera alors choisi comme nouvelle variable.

\subsection 2.2. Cas des \'equations non r\'esolues incompl\`etes|
\medskip
\vbox{
\noindent {\tenpointbf a) \'Equations du type~ $(E)$~: $f(x,y')=0$}
\smallskip
Supposons que l'\'equation $f(x,p)=0$ admette une param\'etrisation de
classe $C^1$}
$$
\cases{
x=\varphi(t)\cr
\noalign{\vskip3pt}
p=\psi(t).\cr}
$$
On a alors
$$
\eqalign{
dx&=\varphi'(t)\,dt\cr
dy&=y'\,dx=\psi(t)\,dx=\psi(t)\varphi'(t)\,dt\cr
}
$$
On en d\'eduit
$$
y=\int^t_{t_0}\psi(u)\varphi'(u)du+\lambda=\rho(t)+\lambda,
$$
ce qui donne une param\'etrisation des courbes int\'egrales~:
$$
\cases{
x=\varphi(t)\hfill\cr
\noalign{\vskip3pt}
y=\rho(t)+\lambda.\hfill\cr
}
$$

\bigskip
\noindent {\tenpointbf b) \'Equations du type~ $(E)$~: $f(y,y')=0$,}

\noindent connaissant une param\'etrisation $\cases{
y=\varphi(t)\hfill\cr
\noalign{\vskip3pt}
y'=\psi(t).\hfill\cr
}$

On obtient $dy=\varphi'(t)dt=\psi(t)dx$, d'o\`u $dx=\dfrac{\varphi'(t)}{\psi(t)}\ dt$.
Les courbes int\'egrales sont param\'etr\'ees par
$$
\cases{
x=\rho(t)+\lambda\hfill\cr
\noalign{\vskip3pt}
y=\varphi(t)\hfill\cr
}
$$
avec\quad $\displaystyle \rho(t)=\int^t_{t_0}\dfrac{\varphi'(u)}{\psi(u)}\ du$.

\subsection 2.3. \'Equations homog\`enes non r\'esolues|

\noindent Ce sont les \'equations pouvant \^etre mises sous la forme
$$
f\Big(\frac{y}{x},y'\Big)=0.\leqno{\rm (E)}
$$
Supposons qu'on  connaisse une param\'etrisation
$$
\cases{
\dfrac{y}{x}=\varphi(t)\hfill\cr
\noalign{\vskip3pt}
y'=\psi(t).\hfill\cr
}
$$
On a alors
$$
\eqalign{
&\quad\kern6pt y=x\varphi(t),\cr
&\cases{
dy=\varphi(t)dx+x\varphi'(t)dt\hfill\cr
\noalign{\vskip3pt}
dy=\psi(t)\,dx,\hfill\cr
}\cr
}
$$
d'o\`u\quad $(\psi(t)-\varphi(t))\,dx=x\varphi'(t)\,dt$.

\medskip
{\parindent=0.8pc
\item{$\bullet$} On a d'une part des solutions singuli\`eres correspondant
aux racines $t_j$ de\break $\psi(t)=\varphi(t)$, donnant des droites
$$
y=x\varphi(t_j).
$$
\item{$\bullet$} D'autre part, pour $t\not=t_j$ on obtient
$$
\frac{dx}{x}=\frac{\varphi'(t)}{\psi(t)-\varphi(t)}\ dt,
$$
ce qui donne par int\'egration de $\varphi'/(\psi-\varphi)$~:
$$
\eqalign{
&\quad\ln\,|x|=\rho(t)+C,\cr
&\cases{
x=\lambda e^{\rho(t)}\hfill\cr
\noalign{\vskip3pt}
y=x\varphi(t)=\lambda \varphi(t)e^{\rho(t)}\hfill\cr
},\qquad\lambda\in\bR.\cr
}
$$
Il est clair sur ces derni\`eres formules que les courbes int\'egrales
se d\'eduisent les unes des autres par les homoth\'eties de centre $O$.
\par}

\medskip
\state Exemple -- |  Soit l'\'equation $x^2(y+3xy')=(y+xy')^3$.\hfill\break
En divisant par $x^3$ on trouve
$$
\frac{y}{x}+3y'=\Big(\frac{y}{x}+y'\Big)^3,
$$
c'est donc une \'equation homog\`ene non r\'esolue en $y'$. On obtient une
param\'etrisation en posant
$$
\cases{
\dfrac{y}{x}+y'=t\hfill\cr
\noalign{\vskip3pt}
\dfrac{y}{x}+3y'=t^3,\hfill\cr
}
$$
d'o\`u
$$
\cases{
\dfrac{y}{x}=\dfrac{1}{2}\,(3t-t^3)\qquad\qquad(*)\hfill\cr
\noalign{\vskip3pt}
y'=\dfrac{1}{2}\,(t^3-t).\hfill\cr}
$$

\noindent En diff\'erentiant $y=\dfrac{1}{2}\,(3t-t^3)x$ on obtient
$$
\eqalign{
dy&=\frac{1}{2}\,(3-3t^2)dt\cdot x+\dfrac{1}{2}\,(3t-t^3)\,dx\cr
&=y'\,dx=\frac{1}{2}\,(t^3-t)\,dx,\cr
}
$$
d'o\`u l'\'equation
$$
\eqalign{
&(t^3-2t)dx=\frac{1}{2}\,(3-3t^2)dt\cdot x,\cr
&\frac{dx}{x}=\frac{3(1-t^2)}{2t(t^2-2)}\ dt.\cr
}
$$
$\bullet$ Solutions singuli\`eres~: $t(t^2-2)=0\Leftrightarrow t=0,\sqrt{2},
-\sqrt{2}$. En rempla\c{c}ant dans $(*)$ on obtient les droites
$$
y=0,\qquad y=\frac{\sqrt{2}}{2}\,x,\qquad y=-\frac{\sqrt{2}}{2}\,x.
$$
$\bullet$ Solution g\'en\'erale~:
$$
\eqalign{
&\frac{1-t^2}{t(t^2-2)}=\frac{1-\dfrac{t^2}{2}-\dfrac{t^2}{2}}{t(t^2-2)}
=-\frac{1}{2t}-\frac{t}{2(t^2-2)},\cr
&\frac{3(1-t^2)}{2t(t^2-2)}\ dt=-\frac{3}{4}\ \frac{dt}{t}-\frac{3}{4}\
\frac{tdt}{t^2-2}.\cr 
}
$$
On en d\'eduit
$$
\eqalign{
&\quad\ln\,|x|=-\frac{3}{4}\,\ln\,|t|-\frac{3}{8}\ \ln\,|t^2-2|+C,\cr
&\cases{
x=\lambda|t|^{-3/4}|t^2-2|^{-3/8}\hfill\cr
\noalign{\vskip3pt}
y=\dfrac{y}{x}\cdot x=\dfrac{\lambda}{2}\ (3t-t^3)|t|^{-3/4}|t^2-2|^{-3/8}.\hfill\cr
}\cr
}
$$

\InsertFig 16.000 77.000 {
1 mm unit
 50.000  35.000 moveto 
[  50.000  35.000   80.000  65.000   20.000  65.000   50.000  35.000  
 80.000   5.000   20.000   5.000   50.000  35.000  ] polygon 
closepath 
  0.900 setgray  fill 0 setgray 
 83.100  48.100 moveto
  0.700 disk
 69.700  50.200 moveto
  0.700 disk
  7.574   5.000 moveto 
 92.426  65.000 lineto 
stroke 
 92.426   5.000 moveto 
  7.574  65.000 lineto 
stroke 
  0.000  35.000 moveto 
100.000   0.000 vector 
 50.000   0.000 moveto 
 70.000  90.000 vector 
gsave 
  5.000   5.000 moveto 
 90.000  60.000 rect  clip  newpath 
 95.077  36.289 moveto 
[  95.077  36.289    93.646  36.303    92.217  36.318    90.790  36.333   
 89.366  36.349    87.945  36.366    86.527  36.383    85.112  36.401   
 83.700  36.421    82.292  36.441    80.889  36.463    79.490  36.485   
 78.095  36.509    76.706  36.535    75.323  36.563    73.946  36.592   
 72.576  36.623    71.214  36.657    69.860  36.694    68.516  36.734   
 67.184  36.778    65.863  36.826    64.557  36.878    63.267  36.937   
 61.997  37.003    60.749  37.077    59.527  37.161    58.337  37.257   
 57.186  37.368    56.214  37.481    55.516  37.575    54.990  37.655   
 54.580  37.724    54.253  37.782    53.987  37.832    53.769  37.873   
 53.589  37.908    53.439  37.936    53.316  37.958    53.215  37.975   
 53.134  37.987    53.073  37.995    53.029  37.999    53.005  38.000   
 53.000  38.000   ] scurve 
 53.000  38.000 moveto 
[  53.000  38.000    53.002  38.000    53.008  38.000    53.019  38.001   
 53.034  38.002    53.055  38.004    53.082  38.008    53.117  38.014   
 53.159  38.022    53.211  38.034    53.275  38.050    53.354  38.073   
 53.452  38.106    53.574  38.151    53.730  38.215    53.935  38.307   
 54.216  38.445    54.628  38.667    55.311  39.067    56.785  40.003   
 58.025  40.830    59.173  41.610    60.269  42.364    61.331  43.100   
 62.371  43.823    63.392  44.535    64.399  45.240    65.395  45.938   
 66.381  46.630    67.360  47.318    68.331  48.001    69.296  48.680   
 70.256  49.356    71.210  50.029    72.161  50.698    73.107  51.366   
 74.050  52.031    74.990  52.694    75.926  53.354    76.859  54.013   
 77.790  54.670    78.718  55.325    79.643  55.978    80.566  56.630   
 81.487  57.281    82.406  57.929    83.322  58.577    84.237  59.223   
 85.149  59.868    86.060  60.511    86.969  61.153    87.875  61.794   
 88.780  62.433    89.683  63.071    90.585  63.708    91.484  64.344   
 92.382  64.978    93.278  65.611   ] scurve 
 95.077  43.178 moveto 
[  95.077  43.178    93.712  43.261    92.352  43.348    90.997  43.437   
 89.647  43.531    88.302  43.628    86.963  43.729    85.631  43.835   
 84.305  43.945    82.986  44.060    81.675  44.180    80.372  44.306   
 79.078  44.438    77.793  44.576    76.518  44.721    75.254  44.874   
 74.002  45.034    72.763  45.201    71.537  45.378    70.326  45.563   
 69.132  45.756    67.956  45.958    66.820  46.163    65.878  46.340   
 65.089  46.491    64.426  46.618    63.866  46.724    63.395  46.810   
 63.002  46.877    62.679  46.928    62.419  46.963    62.222  46.985   
 62.085  46.996    62.012  47.000    62.000  47.000   ] scurve 
 62.000  47.000 moveto 
[  62.000  47.000    62.008  47.000    62.032  47.001    62.073  47.003   
 62.134  47.008    62.216  47.017    62.322  47.031    62.456  47.053   
 62.621  47.084    62.825  47.130    63.074  47.193    63.379  47.282   
 63.756  47.405    64.227  47.576    64.824  47.817    65.603  48.163   
 66.659  48.677    67.824  49.289    68.935  49.907    70.011  50.530   
 71.061  51.157    72.091  51.787    73.106  52.419    74.109  53.053   
 75.101  53.688    76.085  54.324    77.060  54.961    78.030  55.599   
 78.994  56.238    79.953  56.876    80.907  57.515    81.857  58.155   
 82.804  58.794    83.748  59.433    84.688  60.073    85.626  60.712   
 86.562  61.352    87.495  61.991    88.427  62.630    89.356  63.269   
 90.283  63.908    91.209  64.547    92.133  65.186    93.056  65.824   
] scurve 
 95.127  52.165 moveto 
[  95.127  52.165    93.836  52.324    92.552  52.487    91.274  52.656   
 90.003  52.829    88.739  53.008    87.483  53.192    86.234  53.381   
 84.994  53.576    83.762  53.777    82.539  53.983    81.325  54.194   
 80.121  54.409    78.927  54.629    77.744  54.852    76.572  55.077   
 75.507  55.281    74.592  55.456    73.806  55.602    73.133  55.721   
 72.563  55.816    72.086  55.888    71.698  55.940    71.394  55.973   
 71.174  55.992    71.041  55.999    71.000  56.000   ] scurve 
 71.000  56.000 moveto 
[  71.000  56.000    71.013  56.000    71.053  56.001    71.122  56.005   
 71.222  56.013    71.358  56.027    71.534  56.050    71.754  56.084   
 72.026  56.135    72.359  56.207    72.765  56.308    73.260  56.447   
 73.866  56.639    74.615  56.903    75.555  57.269    76.761  57.787   
 77.918  58.325    79.035  58.875    80.123  59.435    81.188  60.004   
 82.235  60.580    83.267  61.162    84.286  61.749    85.294  62.340   
 86.293  62.936    87.285  63.534    88.268  64.136    89.246  64.740   
 90.218  65.346    91.185  65.954   ] scurve 
  4.923  33.711 moveto 
[   4.923  33.711     6.354  33.697     7.783  33.682     9.210  33.667   
 10.634  33.651    12.055  33.634    13.473  33.617    14.888  33.599   
 16.300  33.579    17.708  33.559    19.111  33.537    20.510  33.515   
 21.905  33.491    23.294  33.465    24.677  33.437    26.054  33.408   
 27.424  33.377    28.786  33.343    30.140  33.306    31.484  33.266   
 32.816  33.222    34.137  33.174    35.443  33.122    36.733  33.063   
 38.003  32.997    39.251  32.923    40.473  32.839    41.663  32.743   
 42.814  32.632    43.786  32.519    44.484  32.425    45.010  32.345   
 45.420  32.276    45.747  32.218    46.013  32.168    46.231  32.127   
 46.411  32.092    46.561  32.064    46.684  32.042    46.785  32.025   
 46.866  32.013    46.927  32.005    46.971  32.001    46.995  32.000   
 47.000  32.000   ] scurve 
 47.000  32.000 moveto 
[  47.000  32.000    46.998  32.000    46.992  32.000    46.981  31.999   
 46.966  31.998    46.945  31.996    46.918  31.992    46.883  31.986   
 46.841  31.978    46.789  31.966    46.725  31.950    46.646  31.927   
 46.548  31.894    46.426  31.849    46.270  31.785    46.065  31.693   
 45.784  31.555    45.372  31.333    44.689  30.933    43.215  29.997   
 41.975  29.170    40.827  28.390    39.731  27.636    38.669  26.900   
 37.629  26.177    36.608  25.465    35.601  24.760    34.605  24.062   
 33.619  23.370    32.640  22.682    31.669  21.999    30.704  21.320   
 29.744  20.644    28.790  19.971    27.839  19.302    26.893  18.634   
 25.950  17.969    25.010  17.306    24.074  16.646    23.141  15.987   
 22.210  15.330    21.282  14.675    20.357  14.022    19.434  13.370   
 18.513  12.719    17.594  12.071    16.678  11.423    15.763  10.777   
 14.851  10.132    13.940   9.489    13.031   8.847    12.125   8.206   
 11.220   7.567    10.317   6.929     9.415   6.292     8.516   5.656   
  7.618   5.022     6.722   4.389   ] scurve 
  4.923  26.822 moveto 
[   4.923  26.822     6.288  26.739     7.648  26.652     9.003  26.563   
 10.353  26.469    11.698  26.372    13.037  26.271    14.369  26.165   
 15.695  26.055    17.014  25.940    18.325  25.820    19.628  25.694   
 20.922  25.562    22.207  25.424    23.482  25.279    24.746  25.126   
 25.998  24.966    27.237  24.799    28.463  24.622    29.674  24.437   
 30.868  24.244    32.044  24.042    33.180  23.837    34.122  23.660   
 34.911  23.509    35.574  23.382    36.134  23.276    36.605  23.190   
 36.998  23.123    37.321  23.072    37.581  23.037    37.778  23.015   
 37.915  23.004    37.988  23.000    38.000  23.000   ] scurve 
 38.000  23.000 moveto 
[  38.000  23.000    37.992  23.000    37.968  22.999    37.925  22.997   
 37.863  22.992    37.779  22.983    37.671  22.968    37.534  22.946   
 37.364  22.913    37.155  22.866    36.899  22.799    36.584  22.706   
 36.194  22.578    35.706  22.398    35.082  22.143    34.263  21.774   
 33.141  21.221    31.987  20.608    30.882  19.989    29.812  19.365   
 28.765  18.738    27.738  18.108    26.725  17.475    25.724  16.841   
 24.733  16.206    23.751  15.569    22.777  14.932    21.808  14.294   
 20.845  13.655    19.887  13.017    18.934  12.378    17.984  11.738   
 17.038  11.099    16.094  10.459    15.154   9.820    14.217   9.180   
 13.281   8.541    12.349   7.902    11.418   7.263    10.489   6.623   
  9.561   5.985     8.636   5.346     7.712   4.707     6.789   4.069   
] scurve 
  4.873  17.835 moveto 
[   4.873  17.835     6.164  17.676     7.448  17.513     8.726  17.344   
  9.997  17.171    11.261  16.992    12.517  16.808    13.766  16.619   
 15.006  16.424    16.238  16.223    17.461  16.017    18.675  15.806   
 19.879  15.591    21.073  15.371    22.256  15.148    23.428  14.923   
 24.493  14.719    25.408  14.544    26.194  14.398    26.867  14.279   
 27.437  14.184    27.914  14.112    28.302  14.060    28.606  14.027   
 28.826  14.008    28.959  14.001    29.000  14.000   ] scurve 
 29.000  14.000 moveto 
[  29.000  14.000    28.987  14.000    28.947  13.999    28.878  13.995   
 28.778  13.987    28.642  13.973    28.466  13.950    28.246  13.916   
 27.974  13.865    27.641  13.793    27.235  13.692    26.740  13.553   
 26.134  13.361    25.385  13.097    24.445  12.731    23.239  12.213   
 22.082  11.675    20.965  11.125    19.877  10.565    18.812   9.996   
 17.765   9.420    16.733   8.838    15.714   8.251    14.706   7.660   
 13.707   7.064    12.715   6.466    11.732   5.864    10.754   5.260   
  9.782   4.654     8.815   4.046   ] scurve 
 76.365  16.336 moveto 
[  76.365  16.336    75.532  16.924    74.700  17.510    73.871  18.096   
 73.043  18.679    72.218  19.261    71.395  19.841    70.574  20.420   
 69.755  20.997    68.939  21.571    68.126  22.144    67.315  22.714   
 66.508  23.281    65.703  23.846    64.902  24.407    64.105  24.966   
 63.312  25.520    62.524  26.071    61.740  26.617    60.962  27.157   
 60.189  27.691    59.423  28.218    58.664  28.737    57.914  29.245   
 57.173  29.741    56.442  30.220    55.725  30.678    55.023  31.106   
 54.462  31.424    54.106  31.610    53.856  31.730    53.670  31.811   
 53.528  31.867    53.415  31.907    53.325  31.936    53.251  31.956   
 53.192  31.971    53.143  31.981    53.104  31.989    53.072  31.993   
 53.047  31.997    53.028  31.998    53.014  31.999    53.005  32.000   
 53.001  32.000    53.000  32.000   ] scurve 
 53.000  32.000 moveto 
[  53.000  32.000    53.011  32.000    53.041  32.002    53.090  32.007   
 53.157  32.016    53.244  32.030    53.352  32.048    53.483  32.072   
 53.641  32.102    53.833  32.139    54.064  32.183    54.347  32.235   
 54.697  32.297    55.139  32.368    55.711  32.453    56.481  32.552   
 57.573  32.671    59.253  32.819    60.953  32.936    62.621  33.031   
 64.268  33.109    65.898  33.176    67.517  33.234    69.124  33.284   
 70.724  33.330    72.316  33.370    73.903  33.407    75.484  33.441   
 77.060  33.472    78.632  33.500    80.201  33.526    81.766  33.551   
 83.328  33.574    84.887  33.596    86.444  33.616    87.999  33.635   
 89.551  33.653    91.102  33.670    92.651  33.687    94.198  33.702   
 95.743  33.717   ] scurve 
 92.796   4.729 moveto 
[  92.796   4.729    91.948   5.328    91.102   5.927    90.256   6.524   
 89.410   7.122    88.565   7.719    87.721   8.315    86.878   8.911   
 86.036   9.506    85.194  10.101    84.353  10.695    83.513  11.288   
 82.674  11.881    81.836  12.473    80.999  13.064    80.163  13.655   
 79.328  14.244    78.494  14.833    77.662  15.421    76.830  16.007   
 76.000  16.593    75.172  17.178    74.345  17.761    73.520  18.343   
 72.696  18.924    71.874  19.504    71.054  20.081    70.236  20.658   
 69.421  21.232    68.607  21.805    67.797  22.375    66.989  22.943   
 66.279  23.442    65.667  23.871    65.133  24.246    64.660  24.577   
 64.238  24.873    63.857  25.140    63.511  25.381    63.195  25.602   
 62.905  25.805    62.637  25.991    62.389  26.165    62.215  26.286   
] scurve 
 92.289   4.707 moveto 
[  92.289   4.707    91.435   5.297    90.581   5.886    89.728   6.475   
 88.875   7.062    88.022   7.648    87.169   8.232    86.317   8.816   
 85.465   9.398    84.613   9.978    83.761  10.557    82.910  11.134   
 82.059  11.710    81.208  12.283    80.357  12.854    79.506  13.422   
 78.655  13.988    77.804  14.550    76.953  15.109    76.101  15.665   
 75.250  16.217    74.397  16.764    73.545  17.305    72.691  17.841   
 71.837  18.370    70.981  18.891    70.124  19.403    69.265  19.904   
 68.404  20.392    67.540  20.864    66.673  21.316    65.803  21.743   
 64.974  22.119    64.343  22.379    63.849  22.563    63.454  22.695   
 63.135  22.790    62.874  22.858    62.662  22.907    62.489  22.942   
 62.348  22.965    62.237  22.981    62.150  22.990    62.085  22.996   
 62.039  22.999    62.011  23.000    62.000  23.000    62.000  23.000   
] scurve 
 62.000  23.000 moveto 
[  62.000  23.000    62.035  23.001    62.136  23.007    62.298  23.023   
 62.522  23.050    62.808  23.092    63.160  23.149    63.585  23.224   
 64.092  23.319    64.694  23.433    65.407  23.571    66.257  23.732   
 67.276  23.920    68.511  24.139    69.928  24.374    71.343  24.593   
 72.761  24.798    74.183  24.990    75.608  25.170    77.038  25.339   
 78.472  25.498    79.910  25.647    81.352  25.789    82.798  25.923   
 84.248  26.050    85.700  26.171    87.156  26.285    88.614  26.395   
 90.074  26.499    91.537  26.599    93.003  26.694    94.470  26.786   
 95.939  26.873   ] scurve 
 90.223   4.651 moveto 
[  90.223   4.651    89.341   5.202    88.458   5.748    87.572   6.291   
 86.686   6.828    85.797   7.361    84.905   7.889    84.011   8.410   
 83.115   8.925    82.215   9.431    81.311   9.929    80.403  10.417   
 79.491  10.893    78.573  11.356    77.649  11.804    76.719  12.232   
 75.782  12.637    74.837  13.014    74.044  13.301    73.405  13.509   
 72.883  13.660    72.456  13.770    72.106  13.849    71.819  13.904   
 71.586  13.943    71.400  13.968    71.254  13.984    71.144  13.993   
 71.068  13.998    71.021  14.000    71.001  14.000    71.000  14.000   
] scurve 
 71.000  14.000 moveto 
[  71.000  14.000    71.049  14.001    71.189  14.009    71.415  14.029   
 71.726  14.064    72.122  14.117    72.606  14.191    73.184  14.288   
 73.866  14.409    74.661  14.557    75.588  14.734    76.667  14.942   
 77.925  15.182    79.284  15.437    80.639  15.684    81.992  15.923   
 83.347  16.154    84.703  16.377    86.063  16.592    87.427  16.800   
 88.794  17.000    90.165  17.193    91.540  17.380    92.920  17.560   
 94.303  17.735    95.690  17.903   ] scurve 
 23.793  53.553 moveto 
[  23.793  53.553    24.626  52.965    25.457  52.379    26.286  51.794   
 27.113  51.210    27.938  50.629    28.761  50.049    29.581  49.471   
 30.400  48.894    31.215  48.320    32.028  47.748    32.838  47.179   
 33.645  46.612    34.449  46.048    35.249  45.487    36.045  44.929   
 36.837  44.375    37.625  43.825    38.408  43.281    39.185  42.741   
 39.957  42.208    40.721  41.683    41.478  41.166    42.227  40.660   
 42.966  40.167    43.695  39.691    44.410  39.239    45.108  38.818   
 45.617  38.533    45.948  38.363    46.184  38.252    46.360  38.177   
 46.496  38.124    46.604  38.087    46.691  38.060    46.761  38.041   
 46.819  38.027    46.865  38.017    46.903  38.010    46.933  38.006   
 46.957  38.003    46.975  38.001    46.988  38.000    46.996  38.000   
 47.000  38.000    47.000  38.000   ] scurve 
 47.000  38.000 moveto 
[  47.000  38.000    46.989  38.000    46.959  37.998    46.910  37.993   
 46.843  37.984    46.756  37.970    46.648  37.952    46.517  37.928   
 46.359  37.898    46.167  37.861    45.936  37.817    45.653  37.765   
 45.303  37.703    44.861  37.632    44.289  37.547    43.519  37.448   
 42.427  37.329    40.747  37.181    39.047  37.064    37.379  36.969   
 35.732  36.891    34.102  36.824    32.483  36.766    30.876  36.716   
 29.276  36.670    27.684  36.630    26.097  36.593    24.516  36.559   
 22.940  36.528    21.368  36.500    19.799  36.474    18.234  36.449   
 16.672  36.426    15.113  36.404    13.556  36.384    12.001  36.365   
 10.449  36.347     8.898  36.330     7.349  36.313     5.802  36.298   
  4.257  36.283   ] scurve 
  7.064  65.369 moveto 
[   7.064  65.369     7.912  64.771     8.759  64.172     9.605  63.574   
 10.450  62.977    11.295  62.380    12.139  61.783    12.983  61.187   
 13.825  60.592    14.667  59.997    15.508  59.403    16.348  58.810   
 17.187  58.217    18.026  57.625    18.863  57.033    19.699  56.443   
 20.534  55.853    21.368  55.264    22.201  54.676    23.032  54.090   
 23.862  53.504    24.691  52.919    25.518  52.335    26.344  51.753   
 27.168  51.172    27.990  50.592    28.810  50.014    29.628  49.438   
 30.444  48.863    31.258  48.290    32.069  47.719    32.878  47.151   
 33.608  46.638    34.234  46.198    34.781  45.814    35.264  45.476   
 35.695  45.174    36.082  44.903    36.434  44.657    36.754  44.433   
 37.048  44.228    37.320  44.038    37.571  43.863    37.785  43.714   
] scurve 
  7.711  65.293 moveto 
[   7.711  65.293     8.565  64.703     9.419  64.114    10.272  63.525   
 11.125  62.938    11.978  62.352    12.831  61.768    13.683  61.184   
 14.535  60.602    15.387  60.022    16.239  59.443    17.090  58.866   
 17.941  58.290    18.792  57.717    19.643  57.146    20.494  56.578   
 21.345  56.012    22.196  55.450    23.047  54.891    23.899  54.335   
 24.750  53.783    25.603  53.236    26.455  52.695    27.309  52.159   
 28.163  51.630    29.019  51.109    29.876  50.597    30.735  50.096   
 31.596  49.608    32.460  49.136    33.327  48.684    34.197  48.257   
 35.026  47.881    35.657  47.621    36.151  47.437    36.546  47.305   
 36.865  47.210    37.126  47.142    37.338  47.093    37.511  47.058   
 37.652  47.035    37.763  47.019    37.850  47.010    37.915  47.004   
 37.961  47.001    37.989  47.000    38.000  47.000    38.000  47.000   
] scurve 
 38.000  47.000 moveto 
[  38.000  47.000    37.965  46.999    37.864  46.993    37.702  46.977   
 37.478  46.950    37.192  46.908    36.840  46.851    36.415  46.776   
 35.908  46.681    35.306  46.567    34.593  46.429    33.743  46.268   
 32.724  46.080    31.489  45.861    30.072  45.626    28.657  45.407   
 27.239  45.202    25.817  45.010    24.392  44.830    22.962  44.661   
 21.528  44.502    20.090  44.353    18.648  44.211    17.202  44.077   
 15.752  43.950    14.300  43.829    12.844  43.715    11.386  43.605   
  9.926  43.501     8.463  43.401     6.997  43.306     5.530  43.214   
  4.061  43.127   ] scurve 
  9.777  65.349 moveto 
[   9.777  65.349    10.659  64.798    11.542  64.252    12.428  63.709   
 13.314  63.172    14.203  62.639    15.095  62.111    15.989  61.590   
 16.885  61.075    17.785  60.569    18.689  60.071    19.597  59.583   
 20.509  59.107    21.427  58.644    22.351  58.196    23.281  57.768   
 24.218  57.363    25.163  56.986    25.956  56.699    26.595  56.491   
 27.117  56.340    27.544  56.230    27.894  56.151    28.181  56.096   
 28.414  56.057    28.600  56.032    28.746  56.016    28.856  56.007   
 28.932  56.002    28.979  56.000    28.999  56.000    29.000  56.000   
] scurve 
 29.000  56.000 moveto 
[  29.000  56.000    28.951  55.999    28.811  55.991    28.585  55.971   
 28.274  55.936    27.878  55.883    27.394  55.809    26.816  55.712   
 26.134  55.591    25.339  55.443    24.412  55.266    23.333  55.058   
 22.075  54.818    20.716  54.563    19.361  54.316    18.008  54.077   
 16.653  53.846    15.297  53.623    13.937  53.408    12.573  53.200   
 11.206  53.000     9.835  52.807     8.460  52.620     7.080  52.440   
  5.697  52.265     4.310  52.097   ] scurve 
 90.544  63.251 moveto 
[  93.168  65.149    92.058  64.347
 90.950  63.546    89.844  62.745    88.330  61.645    87.227  60.842   
 86.126  60.040    85.029  59.238    83.935  58.435    82.844  57.633   
 81.757  56.830    80.674  56.027    79.595  55.224    78.520  54.420   
 77.450  53.614    76.385  52.808    75.326  51.999    74.273  51.189   
 73.226  50.376    72.187  49.559    71.156  48.739    70.134  47.913   
 69.121  47.082    68.120  46.242    67.131  45.394    66.156  44.535   
 65.197  43.662    64.255  42.773    63.334  41.864    62.435  40.932   
 61.562  39.972    60.717  38.978    59.906  37.947    59.130  36.872   
 58.395  35.747    57.704  34.568    57.059  33.330    56.462  32.030   
 55.915  30.667    55.418  29.242    54.968  27.755    54.564  26.212   
 54.202  24.616    53.879  22.972    53.592  21.285    53.335  19.561   
 53.107  17.803    52.902  16.017    52.720  14.206    52.555  12.373   
 52.408  10.522    52.274   8.654    52.153   6.772    52.044   4.879   
 51.944   2.974   ] scurve 
 95.131  60.558 moveto 
[  95.131  60.558    94.131  59.620    93.137  58.675    92.151  57.722   
 91.172  56.762    90.201  55.793    89.239  54.817    88.286  53.831   
 87.342  52.835    86.408  51.829    85.484  50.813    84.571  49.785   
 83.669  48.746    82.779  47.694    81.902  46.629    81.037  45.550   
 80.186  44.457    79.348  43.349    78.525  42.226    77.717  41.086   
 76.924  39.930    76.148  38.756    75.387  37.565    74.644  36.355   
 73.918  35.127    73.209  33.880    72.518  32.613    71.845  31.326   
 71.191  30.020    70.555  28.693    69.938  27.346    69.339  25.979   
 68.759  24.591    68.197  23.183    67.654  21.755    67.129  20.308   
 66.622  18.840    66.133  17.354    65.661  15.848    65.207  14.324   
 64.769  12.782    64.347  11.222    63.941   9.645    63.550   8.051   
 63.175   6.441    62.814   4.816    62.466   3.176   ] scurve 
 95.080  43.521 moveto 
[  95.080  43.521    94.290  42.342    93.509  41.152    92.739  39.952   
 91.979  38.741    91.230  37.518    90.491  36.285    89.764  35.040   
 89.046  33.784    88.340  32.516    87.645  31.237    86.962  29.945   
 86.289  28.642    85.628  27.327    84.978  26.000    84.339  24.661   
 83.712  23.310    83.097  21.947    82.493  20.571    81.900  19.184   
 81.319  17.784    80.749  16.373    80.190  14.949    79.643  13.514   
 79.107  12.067    78.582  10.608    78.068   9.138    77.565   7.656   
 77.073   6.162    76.592   4.658    76.121   3.142   ] scurve 
 12.015   8.605 moveto 
[  6.832   4.851     7.942   5.653     9.050   6.454    10.156   7.255
 12.015   8.605    13.118   9.409    14.218  10.212    15.315  11.014   
 16.409  11.817    17.499  12.620    18.585  13.423    19.668  14.227   
 20.746  15.031    21.820  15.836    22.889  16.642    23.952  17.449   
 25.010  18.258    26.061  19.070    27.106  19.884    28.143  20.702   
 29.171  21.524    30.190  22.351    31.199  23.185    32.197  24.027   
 33.182  24.879    34.152  25.743    35.106  26.620    36.042  27.516   
 36.956  28.431    37.847  29.372    38.712  30.342    39.546  31.347   
 40.346  32.392    41.109  33.482    41.831  34.624    42.508  35.822   
 43.138  37.079    43.718  38.399    44.249  39.782    44.731  41.228   
 45.166  42.732    45.556  44.293    45.905  45.905    46.216  47.563   
 46.493  49.263    46.741  50.998    46.961  52.765    47.158  54.560   
 47.335  56.378    47.494  58.217    47.637  60.074    47.766  61.947   
 47.883  63.832    47.989  65.730   ] scurve 
  4.867   9.440 moveto 
[   4.867   9.440     5.867  10.378     6.861  11.324     7.847  12.276   
  8.826  13.237     9.797  14.205    10.759  15.182    11.712  16.168   
 12.656  17.163    13.591  18.169    14.515  19.186    15.428  20.213   
 16.329  21.253    17.219  22.305    18.097  23.370    18.962  24.449   
 19.813  25.542    20.651  26.650    21.474  27.773    22.282  28.913   
 23.075  30.069    23.851  31.242    24.612  32.434    25.355  33.643   
 26.081  34.871    26.790  36.119    27.481  37.385    28.154  38.672   
 28.808  39.979    29.444  41.305    30.062  42.652    30.660  44.020   
 31.241  45.407    31.802  46.815    32.345  48.243    32.870  49.691   
 33.377  51.158    33.866  52.645    34.338  54.150    34.793  55.675   
 35.231  57.217    35.653  58.777    36.059  60.354    36.449  61.947   
 36.825  63.557    37.186  65.182    37.533  66.823   ] scurve 
  4.908  26.461 moveto 
[   4.908  26.461     5.699  27.640     6.479  28.830     7.249  30.030   
  8.009  31.241     8.759  32.463     9.498  33.696    10.226  34.941   
 10.943  36.197    11.649  37.465    12.344  38.744    13.028  40.035   
 13.701  41.338    14.362  42.653    15.012  43.980    15.651  45.319   
 16.278  46.670    16.894  48.033    17.498  49.408    18.091  50.795   
 18.673  52.195    19.243  53.606    19.801  55.029    20.349  56.464   
 20.885  57.911    21.410  59.370    21.924  60.840    22.427  62.322   
 22.919  63.815    23.401  65.320    23.872  66.835   ] scurve 
 52.032  65.337 moveto 
[  52.032  65.337    52.140  63.449    52.259  61.571    52.391  59.705   
 52.537  57.851    52.699  56.011    52.881  54.188    53.084  52.384   
 53.313  50.601    53.572  48.842    53.864  47.111    54.196  45.412   
 54.574  43.748    55.002  42.123    55.488  40.543    56.038  39.009   
 56.657  37.527    57.348  36.096    58.112  34.719    58.949  33.395   
 59.855  32.121    60.823  30.894    61.845  29.710    62.915  28.564   
 64.022  27.453    65.160  26.372    66.323  25.317    67.503  24.284   
 68.698  23.271    69.903  22.275    71.116  21.293    72.334  20.324   
 73.557  19.367    74.782  18.419    76.008  17.479    77.236  16.547   
 78.464  15.623    79.692  14.704    80.920  13.790    82.146  12.882   
 83.373  11.978    84.598  11.079    85.822  10.183    87.044   9.290   
 88.266   8.401    89.486   7.515    90.705   6.632    91.922   5.751   
 93.138   4.873    94.353   3.997   ] scurve 
 62.783  65.326 moveto 
[  62.783  65.326    63.148  63.678    63.528  62.043    63.924  60.423   
 64.336  58.819    64.765  57.230    65.212  55.657    65.677  54.100   
 66.161  52.561    66.663  51.039    67.185  49.535    67.727  48.049   
 68.289  46.581    68.872  45.133    69.476  43.703    70.101  42.292   
 70.747  40.901    71.414  39.529    72.102  38.176    72.812  36.842   
 73.542  35.528    74.293  34.232    75.065  32.955    75.856  31.696   
 76.667  30.454    77.497  29.231    78.346  28.024    79.213  26.834   
 80.097  25.659    80.997  24.500    81.914  23.356    82.846  22.227   
 83.793  21.111    84.753  20.008    85.727  18.917    86.714  17.839   
 87.712  16.772    88.722  15.716    89.742  14.671    90.773  13.635   
 91.813  12.608    92.861  11.591    93.918  10.581    94.669   9.874   
] scurve 
 76.618  65.258 moveto 
[  76.618  65.258    77.107  63.733    77.608  62.218    78.119  60.715   
 78.643  59.222    79.178  57.741    79.724  56.270    80.283  54.812   
 80.854  53.364    81.437  51.928    82.032  50.503    82.640  49.091   
 83.260  47.689    83.892  46.300    84.536  44.922    85.193  43.556   
 85.863  42.201    86.544  40.859    87.239  39.528    87.945  38.208   
 88.663  36.900    89.394  35.603    90.137  34.318    90.891  33.044   
 91.657  31.781    92.435  30.530    93.224  29.289    94.025  28.058   
 94.837  26.839    95.660  25.630   ] scurve 
 47.968   4.663 moveto 
[  47.968   4.663    47.860   6.551    47.741   8.429    47.609  10.295   
 47.463  12.149    47.301  13.989    47.119  15.812    46.916  17.616   
 46.687  19.399    46.428  21.158    46.136  22.889    45.804  24.588   
 45.426  26.252    44.998  27.877    44.512  29.457    43.962  30.991   
 43.343  32.473    42.652  33.904    41.888  35.281    41.051  36.605   
 40.145  37.879    39.177  39.106    38.155  40.290    37.085  41.436   
 35.978  42.547    34.840  43.628    33.677  44.683    32.497  45.716   
 31.302  46.729    30.097  47.725    28.884  48.707    27.666  49.676   
 26.443  50.633    25.218  51.581    23.992  52.521    22.764  53.453   
 21.536  54.377    20.308  55.296    19.080  56.210    17.854  57.118   
 16.627  58.022    15.402  58.921    14.178  59.817    12.956  60.710   
 11.734  61.599    10.514  62.485     9.295  63.368     8.078  64.249   
  6.862  65.127     5.647  66.003   ] scurve 
 37.211   4.704 moveto 
[  37.211   4.704    36.845   6.352    36.465   7.986    36.069   9.606   
 35.656  11.210    35.227  12.799    34.780  14.372    34.314  15.928   
 33.831  17.467    33.328  18.988    32.805  20.492    32.263  21.978   
 31.700  23.445    31.117  24.893    30.513  26.323    29.888  27.733   
 29.242  29.124    28.574  30.496    27.885  31.848    27.175  33.181   
 26.445  34.496    25.693  35.791    24.921  37.068    24.129  38.327   
 23.318  39.568    22.488  40.791    21.639  41.998    20.772  43.188   
 19.887  44.362    18.986  45.520    18.069  46.664    17.137  47.794   
 16.190  48.909    15.229  50.012    14.255  51.102    13.268  52.180   
 12.270  53.247    11.260  54.303    10.239  55.348     9.209  56.384   
  8.169  57.410     7.120  58.428     6.063  59.437     4.998  60.438   
  3.925  61.432   ] scurve 
 23.379   4.750 moveto 
[  23.379   4.750    22.890   6.275    22.390   7.790    21.878   9.293   
 21.355  10.786    20.819  12.267    20.273  13.738    19.714  15.196   
 19.143  16.644    18.560  18.080    17.964  19.504    17.357  20.917   
 16.737  22.318    16.105  23.708    15.460  25.086    14.803  26.452   
 14.134  27.806    13.452  29.149    12.758  30.480    12.051  31.799   
 11.333  33.107    10.602  34.404     9.859  35.689     9.105  36.963   
  8.339  38.225     7.561  39.477     6.771  40.718     5.970  41.948   
  5.159  43.168     4.336  44.377   ] scurve 
[ 0.300   0.600 ] 0 setdash
 95.032  46.770 moveto 
[  95.032  46.770    93.698  46.885    92.371  47.005    91.049  47.130   
 89.733  47.258    88.424  47.392    87.121  47.530    85.825  47.674   
 84.538  47.823    83.258  47.978    81.987  48.139    80.724  48.306   
 79.472  48.480    78.230  48.660    76.998  48.848    75.778  49.042   
 74.571  49.244    73.376  49.451    72.196  49.665    71.031  49.883   
 70.024  50.075    69.169  50.240    68.442  50.378    67.823  50.493   
 67.299  50.585    66.860  50.657    66.499  50.710    66.212  50.746   
 65.996  50.767    65.853  50.777    65.785  50.780    65.780  50.780   
] polygon stroke 
 65.780  50.780 moveto 
[  65.780  50.780    65.790  50.780    65.822  50.781    65.876  50.784   
 65.956  50.791    66.064  50.802    66.204  50.821    66.379  50.849   
 66.597  50.891    66.865  50.951    67.192  51.034    67.594  51.151   
 68.090  51.313    68.708  51.538    69.493  51.855    70.518  52.309   
 71.697  52.880    72.824  53.461    73.914  54.050    74.977  54.647   
 76.020  55.250    77.046  55.857    78.059  56.467    79.061  57.081   
 80.054  57.698    81.038  58.317    82.015  58.938    82.986  59.560   
 83.951  60.184    84.912  60.809    85.868  61.435    86.820  62.062   
 87.768  62.690    88.714  63.318    89.656  63.947    90.596  64.576   
 91.533  65.206    92.468  65.837   ] polygon stroke
 65.072  65.344 moveto 
[  65.072  65.344    65.365  64.130    65.666  62.923    65.976  61.724   
 66.294  60.532    66.620  59.350    66.956  58.175    67.300  57.009   
 67.653  55.851    68.016  54.702    68.388  53.562    68.770  52.432   
 69.162  51.310    69.563  50.197    69.975  49.094    70.396  48.001   
 70.828  46.917    71.270  45.843    71.722  44.778    72.185  43.723   
 72.658  42.678    73.142  41.643    73.636  40.618    74.140  39.602   
 74.655  38.596    75.180  37.600    75.716  36.614    76.262  35.637   
 76.818  34.670    77.384  33.712    77.960  32.763    78.546  31.824   
 79.141  30.894    79.746  29.973    80.360  29.060    80.984  28.156   
 81.616  27.261    82.258  26.374    82.907  25.495    83.566  24.625   
 84.232  23.761    84.906  22.906    85.589  22.058    86.278  21.217   
 86.975  20.383    87.679  19.555    88.390  18.735    89.107  17.920   
 89.831  17.112    90.561  16.310    91.297  15.514    92.039  14.723   
 92.786  13.938    93.539  13.158    94.296  12.383    95.059  11.612   
 95.827  10.847   ] polygon stroke
grestore}
\LabelTeX   98.000  31.200 $x$\ELTX
\LabelTeX   46.500  68.000 $y$\ELTX
\EndFig
\vskip6mm

\majedeb
\state Exercice -- |  {\it Montrer que par tout point $(x,y)$ tel que
$\vert y\vert<\vert x \vert$ il passe exactement trois courbes
int\'egrales, alors qu'il n'en passe qu'une si $\vert y\vert>\vert
x\vert$. Combien en passe-t-il si $\vert y\vert=\vert x\vert$~? {\rm
[Indication~:} \'etudier le nombre de valeurs de $t$ et $y'$ associ\'ees
\`a une valeur donn\'ee de $y/x]$.}

\par\majefin\par

\subsection 2.4. \'Equations de Lagrange (ou \'equations \`a isoclines 
rectilignes)|

\noindent Cherchons \`a d\'eterminer les \'equations diff\'erentielles
dont les courbes isoclines sont des droites. La courbe isocline $y'=p$
sera une droite $y=a(p)x+b(p)$ (pour simplifier, on \'ecarte le cas des
droites parall\`eles \`a $y'Oy$). L'\'equation diff\'erentielle
correspondante est donc
$$
y=a(y')x+b(y').\leqno{\rm (E)}
$$
On supposera que $a,b$ sont au moins de classe $C^1$.

\medskip \state M\'ethode de R\'esolution -- |  On choisit $p=y'$
comme nouvelle variable param\'etrant chaque courbe int\'egrale~; ceci
est l\'egitime \`a condition que $y'$ ne soit pas une constante sur un
morceau de la courbe int\'egrale consid\'er\'ee. Dans le cas
contraire, si $y'=p_0=$ constante, la courbe int\'egrale est contenue
dans la droite $y=a(p_0)x+b(p_0)$, ce qui n'est compatible avec la
condition $y'=p_0$ que si $a(p_0)=p_0$.

\medskip
\noindent $\bullet$ On a donc des solutions singuli\`eres
$y=p_jx+b(p_j)$ o\`u les $p_j$ sont les racines de $a(p)=p$.

\medskip
\noindent$\bullet$ Solution g\'en\'erale~:
$$
\eqalign{
&\quad y=a(p)x+b(p),\cr
&\cases{
dy=a(p)dx+(a'(p)x+b'(p))dp\hfill\cr
\noalign{\vskip3pt}
dy=y'dx=pdx.\hfill\cr
}\cr
}
$$
Il vient
$$
(p-a(p))dx=(a'(p)x+b'(p))dp,
$$
et pour $p\not= a(p)$ on aboutit \`a
$$
\frac{dx}{dp}=\frac{1}{p-a(p)}\ (a'(p)x+b'(p))\,;
$$
c'est une \'equation lin\'eaire en la fonction $x(p)$. La solution
g\'en\'erale sera de la forme
$$
\cases{
x(p)=x_{(1)}(p)+\lambda z(p),\quad\lambda\in \bR,\hfill\cr
\noalign{\vskip3pt}
y(p)=a(p)(x_{(1)}(p)+\lambda z(p))+b(p).\hfill\cr
}
$$

\medskip
\majedeb
\state Exercice -- |  {\it R\'esoudre l'\'equation $2y-x(y'+y'^3)+y'^2=0$.}

\par\majefin\par

\subsection 2.5. \'Equations de Clairaut|

\noindent C'est le cas particulier des \'equations de Lagrange dans lequel $a(p)=p$ pour
toute valeur de $p$, soit
$$
y=y'x+b(y').\leqno{\rm (E)}
$$
Les droites
$$
D_p:y=px+b(p)
$$
qui \'etaient pr\'ec\'edemment des solutions singuli\`eres forment maintenant une famille
g\'en\'erale de solutions.

Montrons que les droites $D_p$ poss\`edent toujours une enveloppe
$\Gamma$. Une telle courbe $\Gamma$ admet par d\'efinition une
param\'etrisation $(x(p),y(p))$ telle que $\Gamma$ soit tangente \`a
$D_p$ au point $(x(p),y(p))$.

\InsertFig -24.000 92.000 {
1 mm unit
gsave 
 63.000  50.000 moveto 
 67.000  45.000 rect  clip  newpath 
 68.982  49.927 moveto 
[  68.982  49.927    69.215  51.184    69.458  52.430    69.712  53.667   
 69.976  54.892    70.251  56.105    70.539  57.306    70.838  58.495   
 71.150  59.671    71.476  60.832    71.816  61.979    72.171  63.111   
 72.542  64.226    72.929  65.324    73.334  66.404    73.758  67.464   
 74.202  68.504    74.667  69.522    75.155  70.516    75.667  71.485   
 76.206  72.426    76.773  73.338    77.371  74.218    78.003  75.062   
 78.672  75.868    79.382  76.631    80.137  77.347    80.943  78.010   
 81.807  78.612    82.736  79.145    83.741  79.598    84.836  79.957   
 86.037  80.204    87.368  80.311    88.865  80.241    90.236  80.017   
 91.484  79.689    92.645  79.288    93.738  78.828    94.779  78.322   
 95.776  77.778    96.736  77.202    97.665  76.598    98.567  75.969   
 99.445  75.319   100.302  74.649   101.140  73.962   101.960  73.260   
102.765  72.543   103.556  71.813   104.333  71.071   105.098  70.318   
105.853  69.555   106.596  68.782   107.330  67.999   108.055  67.209   
108.771  66.410   109.479  65.604   110.179  64.791   110.872  63.971   
111.559  63.144   112.238  62.312   112.912  61.473   113.580  60.630   
114.242  59.781   114.899  58.926   115.550  58.068   116.197  57.204   
116.839  56.336   117.476  55.464   118.110  54.588   118.739  53.707   
119.363  52.823   119.985  51.935   120.602  51.044   121.215  50.149   
121.826  49.251   ] scurve 
 75.155  70.516 moveto 
  0.700 disk
 83.655  85.516 moveto 
 66.655  55.516 lineto 
stroke 
122.330  53.000 moveto 
 92.330  83.000 lineto 
stroke 
110.236  75.017 moveto 
 70.236  85.017 lineto 
stroke 
grestore
gsave 
1 mm unit /nbpt 10 def
 95.000  53.000 moveto 
 57.000  90.000 vector 
 50.000  80.000 moveto 
 90.000   0.000 vector 
 97.400  96.000 moveto 
 75.964 rotmem 
  1.000 nbpt mul it  (y = mx) show 
rotrev 
113.750  70.000 moveto 
-14.036 rotmem 
  1.000 nbpt mul it  (y = ) show currentpoint 
  1.000 nbpt mul symb  (- ) show currentpoint 
  0.065 nbpt mul   0.200 nbpt mul rmoveto 
  1.000 nbpt mul rm  (1) show currentpoint 
3 index 3 index moveto 
  0.000  -0.175 nbpt mul rmoveto 
  1.000 nbpt mul it  (m) show currentpoint 
1 index 5 index moveto 
  1.000 nbpt mul it  ( x) show 
 8.500  1.000 moveto 
 11.000  1.000 lineto
stroke 
rotrev
 53.000  53.000 moveto 
 84.000  54.000 rect  clip  newpath 
[ 2.000  1.000  0.500  1.000 ] 0 setdash
 95.000  80.000 moveto 
 46.000 -14.036 vector
 46.000 165.964 vector  
 46.000  75.964 vector
 46.000 255.964 vector 
[ ] 0 setdash
105.940 109.942 moveto 
[ 105.940 109.942   105.768 108.678   105.607 107.406   105.459 106.124   
105.326 104.832   105.211 103.528   105.116 102.208   105.046 100.870   
105.006  99.510   105.004  98.121   105.049  96.735   105.144  95.398   
105.294  94.113   105.500  92.882   105.767  91.709   106.094  90.593   
106.484  89.535   106.935  88.534   107.447  87.590   108.017  86.699   
108.642  85.860   109.320  85.069   110.045  84.323   110.815  83.617   
111.625  82.949   112.472  82.314   113.352  81.710   114.260  81.134   
115.196  80.582   116.154  80.051   117.134  79.541   118.132  79.048   
119.147  78.570   120.177  78.107   121.220  77.656   122.276  77.217   
123.342  76.788   124.418  76.368   125.502  75.957   126.595  75.553   
127.695  75.156   128.801  74.765   129.913  74.380   131.031  74.000   
132.154  73.625   133.281  73.254   134.413  72.888   135.548  72.524   
136.687  72.165   137.829  71.808   138.974  71.455   140.122  71.104   
141.272  70.756   ] scurve 
 49.997  88.860 moveto 
[  49.997  88.860    51.147  88.508    52.294  88.154    53.437  87.796   
 54.578  87.435    55.716  87.071    56.849  86.703    57.979  86.331   
 59.104  85.954    60.225  85.572    61.340  85.185    62.450  84.792   
 63.554  84.393    64.650  83.986    65.740  83.571    66.820  83.148   
 67.892  82.714    68.953  82.270    70.002  81.814    71.039  81.344   
 72.062  80.859    73.068  80.356    74.056  79.835    75.025  79.292   
 75.970  78.724    76.889  78.129    77.780  77.503    78.638  76.841   
 79.459  76.140    80.240  75.395    80.974  74.600    81.659  73.751   
 82.288  72.842    82.858  71.868    83.364  70.825    83.803  69.709   
 84.173  68.519    84.472  67.254    84.702  65.914    84.865  64.503   
 84.962  63.024    84.999  61.483    84.980  59.884    84.916  58.329   
 84.817  56.821    84.691  55.349    84.544  53.907    84.380  52.489   
 84.201  51.090    84.010  49.709   ] scurve 
116.250  80.000 moveto 
  0.700 disk 
 73.750  80.000 moveto 
  0.700 disk 
grestore}
\LabelTeX   76.500  69.500 $(x(p),y(p))$\ELTX
\LabelTeX   71.236  54.000 $\Gamma$\ELTX
\LabelTeX   64.200  61.000 $D_p$\ELTX
\EndFig
\vskip-41mm

\noindent Le vecteur tangent $(x'(p),y'(p))$ \`a $\Gamma$ doit avoir
m\^eme pente $p$ que $D_p$, d'o\`u $y'(p)=px'(p)$. Par ailleurs
$(x(p),y(p))\in D_p$, donc
$$
y(p)=px(p)+b(p).
$$
En diff\'erentiant, il vient
$$
y'(p)=px'(p)+x(p)+b'(p).
$$
Ceci implique $x(p)+b'(p)=0$, d'o\`u la param\'etrisation cherch\'ee de l'enveloppe~:
$$
\Gamma\ \cases{
x(p)=-b'(p)\hfill\cr
\noalign{\vskip3pt}
y(p)=-pb'(p)+b(p).\hfill\cr
}
$$
Si $b$ est de classe $C^2$, on a $y'(p)=-pb''(p)=px'(p)$ de sorte que $\Gamma$ est bien
l'enveloppe des droites $D_p$. La courbe $\Gamma$ est une solution singuli\`ere de (E).

\bigskip
\majedeb
\state Exercice -- |  {\it R\'esoudre l'\'equation $(xy'-y)(1+y'^2)+1=0$.}

\par\majefin\par

\section 3. Probl\`emes g\'eom\'etriques conduisant \`a des \'equa\-tions 
diff\'erentielles du premier ordre|

\subsection 3.1. \'Equation diff\'erentielle associ\'ee \`a une famille 
de courbes|

\noindent On consid\`ere le probl\`eme suivant~:

\bigskip
\state Probl\`eme -- |  Etant donn\'e une famille de courbes
$$
C_{\lambda}:h(x,y,\lambda)=0,\quad \lambda\in\bR,
$$
existe-t-il une \'equation diff\'erentielle du premier ordre dont les courbes
$C_{\lambda}$ soient les courbes int\'egrales~?

\bigskip
\noindent$\bullet$ {\bf Cas particulier.} On suppose que les courbes 
$C_{\lambda}$ sont les lignes de niveau d'une fonction $V$ de classe $C^1$~:
$$
C_{\lambda}:V(x,y)=\lambda,\qquad\lambda\in\bR.
$$
Alors les courbes $C_{\lambda}$ sont solutions de l'\'equation diff\'erentielle
$$
V'_x(x,y)dx+V'_y(x,y)dy=0.\leqno{\rm (E)}
$$

\bigskip
\noindent$\bullet$ {\bf Cas g\'en\'eral.}
Si l'\'equation $h(x,y,\lambda)=0$ peut se mettre sous la forme 
$\lambda=V(x,y)$, on est ramen\'e au cas pr\'ec\'edent. Sinon on \'ecrit 
que sur chaque $C_{\lambda}$ on a
$$
\cases{
h(x,y,\lambda)=0\hfill\cr
\noalign{\vskip3pt}
h'_x(x,y,\lambda)dx+h'_y(x,y,\lambda)dy=0,\hfill\cr
}
$$
et on essaie d'\'eliminer $\lambda$ entre les 2 \'equations pour
obtenir une \'equation ne faisant plus intervenir que $x,y,dx,dy$.

\medskip
\state Exemple -- |  Soit $C_{\lambda}$ la famille des hyperboles
\'equilat\`eres de centre $O$ passant par le point $A(1,0)$.

\vbox{
\InsertFig -41.000 117.000 {
1 mm unit /nbpt 10 def
 95.000  53.000 moveto 
 57.000  90.000 vector 
 50.000  80.000 moveto 
 90.000   0.000 vector 
 97.400  96.000 moveto 
 75.964 rotmem 
  1.000 nbpt mul it  (y = mx) show 
rotrev 
113.750  70.000 moveto 
-14.036 rotmem 
  1.000 nbpt mul it  (y = ) show currentpoint 
  1.000 nbpt mul symb  (- ) show currentpoint 
  0.065 nbpt mul   0.200 nbpt mul rmoveto 
  1.000 nbpt mul rm  (1) show currentpoint 
3 index 3 index moveto 
  0.000  -0.175 nbpt mul rmoveto 
  1.000 nbpt mul it  (m) show currentpoint 
1 index 5 index moveto 
  1.000 nbpt mul it  ( x) show 
 8.500  1.000 moveto 
 11.000  1.000 lineto
stroke 
rotrev
 53.000  53.000 moveto 
 84.000  54.000 rect  clip  newpath 
[ 2.000  1.000  0.500  1.000 ] 0 setdash
 95.000  80.000 moveto 
 46.000 -14.036 vector
 46.000 165.964 vector  
 46.000  75.964 vector
 46.000 255.964 vector 
[ ] 0 setdash
105.940 109.942 moveto 
[ 105.940 109.942   105.768 108.678   105.607 107.406   105.459 106.124   
105.326 104.832   105.211 103.528   105.116 102.208   105.046 100.870   
105.006  99.510   105.004  98.121   105.049  96.735   105.144  95.398   
105.294  94.113   105.500  92.882   105.767  91.709   106.094  90.593   
106.484  89.535   106.935  88.534   107.447  87.590   108.017  86.699   
108.642  85.860   109.320  85.069   110.045  84.323   110.815  83.617   
111.625  82.949   112.472  82.314   113.352  81.710   114.260  81.134   
115.196  80.582   116.154  80.051   117.134  79.541   118.132  79.048   
119.147  78.570   120.177  78.107   121.220  77.656   122.276  77.217   
123.342  76.788   124.418  76.368   125.502  75.957   126.595  75.553   
127.695  75.156   128.801  74.765   129.913  74.380   131.031  74.000   
132.154  73.625   133.281  73.254   134.413  72.888   135.548  72.524   
136.687  72.165   137.829  71.808   138.974  71.455   140.122  71.104   
141.272  70.756   ] scurve 
 49.997  88.860 moveto 
[  49.997  88.860    51.147  88.508    52.294  88.154    53.437  87.796   
 54.578  87.435    55.716  87.071    56.849  86.703    57.979  86.331   
 59.104  85.954    60.225  85.572    61.340  85.185    62.450  84.792   
 63.554  84.393    64.650  83.986    65.740  83.571    66.820  83.148   
 67.892  82.714    68.953  82.270    70.002  81.814    71.039  81.344   
 72.062  80.859    73.068  80.356    74.056  79.835    75.025  79.292   
 75.970  78.724    76.889  78.129    77.780  77.503    78.638  76.841   
 79.459  76.140    80.240  75.395    80.974  74.600    81.659  73.751   
 82.288  72.842    82.858  71.868    83.364  70.825    83.803  69.709   
 84.173  68.519    84.472  67.254    84.702  65.914    84.865  64.503   
 84.962  63.024    84.999  61.483    84.980  59.884    84.916  58.329   
 84.817  56.821    84.691  55.349    84.544  53.907    84.380  52.489   
 84.201  51.090    84.010  49.709   ] scurve 
116.250  80.000 moveto 
  0.700 disk 
 73.750  80.000 moveto 
  0.700 disk}
\LabelTeX   60.000  76.000 $A'~(-1,0)$\ELTX
\LabelTeX   91.000  76.000 $0$\ELTX
\LabelTeX  138.000  76.000 $x$\ELTX
\LabelTeX   91.500 107.000 $y$\ELTX
\LabelTeX  117.000  82.000 $A~(1,0)$\ELTX
\LabelTeX  106.000 100.000 $C_\lambda$\ELTX
\EndFig
\vskip-44mm}

\noindent Les asymptotes de $C_{\lambda}$ sont alors des droites
orthogonales passant par $O$, soit
$$
y=mx,\qquad y=-\frac{1}{m}\ x,\qquad m\in\bR^*.
$$
Posons $X=y-mx$, $Y=y+\dfrac{1}{m}\ x$. L'\'equation de l'hyperbole
cherch\'ee s'\'ecrit
$$
\eqalign{
&XY=C\quad\hbox{(constante)},\cr
&(y-mx)(y+\frac{1}{m}\,x\Big)=C,\cr
&y^2-x^2+\Big(\frac{1}{m}-m\Big)xy=C.\cr
}$$
En faisant $x=1$, $y=0$ on trouve $C=-1$, d'o\`u l'\'equation
$$
C_{\lambda}:y^2-x^2+\lambda xy+1=0,\qquad \lambda\in\bR,
$$
avec $\lambda=\dfrac{1}{m}-m$ (noter que $m\mapsto\dfrac{1}{m}-m$ est surjective de
$\bR^*$ sur $\bR$). Sur $C_{\lambda}$ on a~:
$$
\eqalign{
\lambda&=\frac{x^2-y^2-1}{xy},\cr
d\lambda&=0=\frac{(2xdx-2ydy)xy-(x^2-y^2-1)(xdy+ydx)}{x^2y^2}.
}$$
L'\'equation diff\'erentielle des courbes $C_{\lambda}$ est donc
$$
\eqalign{
&{\rm (E)}~:\quad(2x^2y-x^2y+y^3+y)dx+(-2xy^2-x^3+xy^2+x)dy=0,\cr
&{\rm (E)}~:\quad(x^2+y^2+1)ydx-(x^2+y^2-1)xdy=0.\cr
}
$$

\subsection 3.2. Recherche des trajectoires orthogonales \`a une famille 
de courbes|

\noindent Soient $(C_{\lambda})$, $(\Gamma_{\mu})$ deux familles de courbes.

\bigskip
\majdeb
\state D\'efinition -- |  {\it On dit que $C_{\lambda}$ et
$\Gamma_{\lambda}$ sont orthogonales si les tangentes \`a $C_{\lambda}$
et $\Gamma_{\mu}$ sont orthogonales en tout point  de
$C_{\lambda}\cap\Gamma_{\mu}$, quels que soient $\lambda$ et $\mu$.}  
\par\majfin\par

\vbox{
\InsertFig -32.000 97.000 {
1 mm unit
 76.700  67.600 moveto 
 78.700  67.800 lineto 
stroke 
 78.700  67.800 moveto 
 78.900  65.800 lineto 
stroke 
gsave 
 60.000  50.000 moveto 
 70.000  40.000 rect  clip  newpath 
 76.436  70.000 moveto 
[  76.436  70.000    76.626  68.583    76.773  67.400    76.905  66.252   
 77.024  65.137    77.130  64.053    77.224  62.998    77.306  61.972   
 77.378  60.974    77.440  60.001    77.492  59.054    77.536  58.131   
 77.572  57.231    77.599  56.354    77.620  55.498    77.633  54.663   
 77.640  53.848    77.641  53.053    77.636  52.277    77.626  51.518   
 77.611  50.777    77.591  50.053    77.567  49.346   ] scurve 
 76.436  70.000 moveto 
[  76.436  70.000    76.223  71.466    76.021  72.773    75.799  74.123   
 75.556  75.517    75.291  76.958    75.046  78.229    74.799  79.453   
 74.536  80.713    74.253  82.010    73.952  83.346    73.630  84.723   
 73.285  86.143    72.918  87.608    72.525  89.122    72.106  90.686   
] scurve 
 91.666  65.496 moveto 
[  91.666  65.496    91.465  63.822    91.261  62.323    91.042  60.879   
 90.810  59.486    90.566  58.142    90.311  56.844    90.047  55.590   
 89.774  54.378    89.494  53.206    89.207  52.072    88.914  50.973   
 88.616  49.910    88.314  48.880   ] scurve 
 91.666  65.496 moveto 
[  91.666  65.496    91.833  67.089    91.954  68.433    92.062  69.820   
 92.155  71.255    92.232  72.739    92.286  74.106    92.323  75.370   
 92.346  76.672    92.356  78.013    92.352  79.395    92.332  80.820   
 92.295  82.291    92.244  83.755    92.188  85.006    92.119  86.292   
 92.037  87.614    91.940  88.974    91.829  90.373   ] scurve 
105.947  65.496 moveto 
[ 105.947  65.496   105.362  63.358   104.750  61.327   104.115  59.393   
103.459  57.550   102.786  55.791   102.098  54.110   101.397  52.503   
100.685  50.965    99.964  49.491   ] scurve 
105.947  65.496 moveto 
[ 105.947  65.496   106.425  67.412   106.850  69.288   107.252  71.249   
107.582  73.037   107.866  74.742   108.130  76.516   108.374  78.362   
108.556  79.931   108.719  81.520   108.866  83.165   108.994  84.871   
109.095  86.478   109.170  87.939   109.231  89.447   109.277  91.004   
] scurve 
120.229  65.496 moveto 
[ 120.229  65.496   119.524  63.473   118.792  61.547   118.036  59.711   
117.257  57.957   116.457  56.279   115.638  54.673   114.802  53.134   
113.949  51.657   113.083  50.239   112.203  48.876   ] scurve 
120.229  65.496 moveto 
[ 120.229  65.496   120.906  67.621   121.552  69.862   122.121  72.051   
122.587  74.035   123.029  76.114   123.407  78.095   123.720  79.915   
124.014  81.812   124.288  83.794   124.495  85.482   124.683  87.195   
124.854  88.975   125.010  90.825   ] scurve 
  0.340 setlinewidth
127.370  52.643 moveto 
[ 127.370  52.643   126.222  53.315   125.297  53.845   124.350  54.375   
123.381  54.906   122.390  55.436   121.375  55.966   120.336  56.494   
119.273  57.020   118.185  57.543   117.071  58.061   115.931  58.576   
114.764  59.084   113.569  59.585   112.414  60.052   111.416  60.441   
110.400  60.823   109.365  61.198   108.311  61.565   107.237  61.924   
106.144  62.274   105.030  62.613   103.897  62.941   102.744  63.257   
101.570  63.560   100.375  63.850    99.160  64.124    97.924  64.383   
 96.667  64.624    95.389  64.847    94.280  65.022    93.228  65.173   
 92.163  65.309    91.085  65.431    89.994  65.538    88.890  65.629   
 87.773  65.703    86.644  65.760    85.502  65.798    84.348  65.818   
 83.182  65.818    82.004  65.797    80.815  65.754    79.615  65.690   
 78.405  65.602    77.185  65.490    75.956  65.353    74.717  65.191   
 73.471  65.002    72.216  64.785    70.955  64.541    69.688  64.266   
 68.415  63.962    67.138  63.627    65.858  63.260    64.574  62.860   
 63.290  62.427    62.005  61.959    60.721  61.457    59.439  60.918   
] scurve 
127.370  52.643 moveto 
[ 127.370  52.643   128.546  51.938   129.609  51.286   130.642  50.639   
] scurve 
 98.807  74.065 moveto 
[  98.807  74.065    97.299  74.200    95.982  74.300    94.648  74.383   
 93.298  74.448    91.932  74.495    90.549  74.521    89.150  74.527   
 87.735  74.511    86.305  74.472    84.860  74.408    83.399  74.319   
 81.924  74.203    80.436  74.058    78.934  73.885    77.419  73.681   
 75.892  73.445    74.354  73.175    72.806  72.872    71.248  72.532   
 69.682  72.155    68.108  71.739    66.529  71.283    64.945  70.787   
 63.357  70.248    61.768  69.665    60.178  69.037    58.590  68.363   
] scurve 
 98.807  74.065 moveto 
[  98.807  74.065   100.294  73.911   101.557  73.763   102.804  73.602   
104.034  73.430   105.248  73.246   106.446  73.052   107.628  72.849   
108.794  72.637   109.944  72.417   111.078  72.189   112.198  71.954   
113.302  71.714   114.391  71.467   115.465  71.216   116.525  70.960   
117.571  70.700   118.602  70.437   119.620  70.170   120.625  69.901   
121.616  69.630   122.594  69.357   123.560  69.083   124.512  68.808   
125.453  68.532   126.382  68.256   127.299  67.980   128.204  67.705   
129.098  67.430   129.981  67.156   130.854  66.883   ] scurve 
 98.807  84.776 moveto 
[  98.807  84.776    97.164  84.775    95.733  84.757    94.288  84.723   
 92.828  84.672    91.355  84.601    89.867  84.512    88.365  84.402   
 86.850  84.270    85.322  84.116    83.780  83.939    82.226  83.736   
 80.660  83.508    79.083  83.253    77.494  82.969    75.894  82.657   
 74.285  82.313    72.666  81.938    71.039  81.531    69.404  81.089   
 67.763  80.612    66.117  80.098    64.465  79.547    62.811  78.957   
 61.154  78.327    59.496  77.656   ] scurve 
 98.807  84.776 moveto 
[  98.807  84.776   100.432  84.758   101.816  84.728   103.186  84.685   
104.541  84.630   105.883  84.563   107.210  84.487   108.523  84.400   
109.822  84.305   111.107  84.201   112.379  84.089   113.637  83.971   
114.882  83.846   116.113  83.715   117.332  83.578   118.538  83.438   
119.732  83.292   120.913  83.143   122.082  82.991   123.240  82.836   
124.386  82.679   125.520  82.519   126.644  82.358   127.756  82.196   
128.858  82.033   129.950  81.870   131.032  81.707   ] scurve 
grestore}
\LabelTeX   56.000  59.000 $C_\lambda$\ELTX
\LabelTeX   73.000  51.000 $\Gamma_\mu$\ELTX
\EndFig
\vskip-34mm}

\majdeb
\state Probl\`eme -- |  {\it Etant donn\'e une famille de courbes
$C_{\lambda}$, trouver la famille $(\Gamma_{\mu})$ des courbes qui sont
orthogonales aux $C_{\lambda}$.} \par\majfin\par

\bigskip
Pour cela, on suppose que l'on conna\^{\i}t une \'equation diff\'erentielle
(E) satisfaite par les courbes $C_{\lambda}$, et on cherche
l'\'equation diff\'erentielle (E$^{\perp}$) des courbes 
orthogonales~$\Gamma_{\mu}$. Distinguons quelques cas.

\medskip
\vbox{
\noindent$\bullet$ $(C_{\lambda})$ satisfait (E)~: $y'=f(x,y)$.

En un point $(x,y)$ donn\'e, la pente de la tangente \`a $C_{\lambda}$
est $y'=f(x,y)$.  La pente de la tangente \`a $\Gamma_{\mu}$ est donc
$-1/f(x,y)$. Les courbes $(\Gamma_{\mu})$ sont donc solutions de
$$
({\rm E}^{\perp})~:\qquad y'=-\frac{1}{f(x,y)}.
$$}
\medskip

$\bullet$ $(C_{\lambda})$ satisfait (E)~: $\displaystyle
\dfrac{d\overrightarrow M}{dt}
=\overrightarrow V(M)\Leftrightarrow
\left\{\eqalign{
\dfrac{dx}{dt}&=a(x,y)\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=b(x,y)\cr}\right.$.

La tangente \`a $C_{\lambda}$ est port\'ee par $\overrightarrow V(M)$,
celle de $(\Gamma_{\mu})$ est donc port\'ee par le vecteur orthogonal
$\overrightarrow V(M)^{\perp}\pmatrix{ -b(x,y)\cr a(x,y)\cr }$. Par
suite $(\Gamma_{\mu})$ est solution de
$$
({\rm E}^{\perp})\quad\left\{\eqalign{
\dfrac{dx}{dt}&=-b(x,y)\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=a(x,y)\cr}\right.
$$
$\bullet$ $(C_{\lambda})$ satisfait (E)~: $\alpha(x,y)dx+\beta(x,y)dy=0$.

\noindent Alors $(\Gamma_{\mu})$ v\'erifie ({\rm E}$^{\perp}$)~:
$-\beta(x,y)dx+\alpha(x,y)dy=0$.

\medskip
\noindent{\bf Cas particulier.} Supposons que les courbes $C_{\lambda}$ sont
les lignes de niveau $V(x,y)=\lambda$ de la fonction $V$. Elles v\'erifient
alors
$$
V'_x(x,y)dx+V'_y(x,y)dy=0.\leqno{\rm (E)}
$$
Leurs trajectoires orthogonales $(\Gamma_{\mu})$ sont les lignes de champ du
gradient $\overrightarrow{\rm grad}\,V$~: \vskip6pt
$({\rm E}^{\perp})\quad\left\{\eqalign{
\dfrac{dx}{dt}&=V'_x(x,y)\cr
\noalign{\vskip3pt}
\dfrac{dy}{dt}&=V'_y(x,y)\cr}\right.$
\bigskip

\state Exemple -- |  Soit $C_{\lambda}:y^2-x^2+\lambda xy+1=0$\quad
(cf.\ \S\ 3.1).

Nous avons vu que $C_{\lambda}$ v\'erifie
$$
{\rm(E)}~:\qquad(x^2+y^2+1)ydx-(x^2+y^2-1)xdy=0.
$$
Donc $\Gamma_{\mu}$ v\'erifie
$$
\eqalign{
({\rm E}^{\perp})~:&\quad\enspace(x^2+y^2-1)xdx+(x^2+y^2+1)ydy=0\cr
&\Leftrightarrow (x^2+y^2)(xdx+ydy)-xdx+ydy=0.\cr
}
$$
Une int\'egrale premi\`ere appara\^{\i}t imm\'ediatement~:
$$
d\Big[\frac{1}{4}\ (x^2+y^2)^2-\frac{x^2}{2}+\frac{y^2}{2}\Big]=0
$$
Les courbes $\Gamma_{\mu}$ sont donc les lignes de niveau
$$
(x^2+y^2)^2-2x^2+2y^2=\mu,\qquad \mu\in\bR,
$$
ce qui peut encore s'\'ecrire
$$
\eqalign{
&(x^2+y^2+1)^2-4x^2=\mu+1,\cr
&(x^2-2x+1+y^2)(x^2+2x+1+y^2)=\mu+1,\cr
&((x-1)^2+y^2)((x+1)^2+y^2)=\mu+1,\cr
&MA\cdot MA'=C=\sqrt{\mu+1},\cr
}
$$
avec
$$
M\pmatrix{x\cr y\cr},\qquad A\pmatrix{A\cr 0\cr},\qquad A'\pmatrix{-1\cr 0\cr}.
$$
Les courbes $MA\cdot MA'=C$ s'appellent des ovales de Cassini. Leur allure
est la suivante.

\vbox{
\InsertFig -34.000 128.000 {
1 mm unit
gsave 
 50.000  50.000 moveto 
100.000  70.000 rect  clip  newpath 
 50.000  85.000 moveto 
100.000   0.000 vector 
100.000  50.000 moveto 
 70.000  90.000 vector 
100.011  84.989 moveto 
[ 100.011  84.989   100.305  84.695   100.753  84.248   101.307  83.699   
101.922  83.095   102.577  82.465   103.262  81.822   103.972  81.176   
104.708  80.534   105.468  79.901   106.255  79.284   107.069  78.685   
107.913  78.110   108.790  77.564   109.703  77.050   110.656  76.575   
111.653  76.144   112.701  75.765   113.805  75.446   114.977  75.200   
116.228  75.044   117.577  75.002   119.001  75.109   120.284  75.348   
121.449  75.696   122.513  76.139   123.487  76.666   124.376  77.273   
125.185  77.955   125.913  78.712   126.560  79.544   127.122  80.454   
127.591  81.447   127.956  82.533   128.198  83.728   128.284  85.054   
128.158  86.540   127.856  87.813   127.426  88.937   126.894  89.943   
126.271  90.850   125.567  91.667   124.785  92.399   123.927  93.050   
122.992  93.618   121.977  94.100   120.876  94.492   119.681  94.782   
118.378  94.957   116.946  94.995   115.440  94.870   114.084  94.621   
112.834  94.278   111.665  93.861   110.562  93.381   109.513  92.848   
108.510  92.267   107.544  91.644   106.610  90.983   105.701  90.285   
104.811  89.554   103.931  88.787   103.051  87.983   102.152  87.127   
101.172  86.168   100.011  85.011   ] scurve 
 99.934  85.066 moveto 
[  99.934  85.066    99.610  85.390    99.134  85.864    98.564  86.429   
 97.939  87.039    97.277  87.674    96.587  88.318    95.870  88.964   
 95.130  89.604    94.364  90.234    93.572  90.848    92.751  91.441   
 91.900  92.010    91.016  92.550    90.094  93.056    89.132  93.522   
 88.124  93.943    87.065  94.310    85.947  94.614    84.760  94.842   
 83.489  94.976    82.115  94.988    80.723  94.852    79.467  94.589   
 78.323  94.220    77.278  93.759    76.322  93.214    75.450  92.592   
 74.658  91.894    73.946  91.122    73.317  90.275    72.773  89.348   
 72.324  88.336    71.983  87.229    71.770  86.010    71.722  84.651   
 71.887  83.213    72.215  81.971    72.664  80.870    73.214  79.883   
 73.852  78.993    74.571  78.192    75.367  77.475    76.239  76.840   
 77.188  76.287    78.218  75.821    79.336  75.447    80.550  75.177   
 81.875  75.025    83.334  75.016    84.808  75.166    86.144  75.433   
 87.378  75.791    88.534  76.220    89.627  76.709    90.667  77.251   
 91.663  77.840    92.623  78.470    93.552  79.138    94.457  79.841   
 95.345  80.579    96.224  81.352    97.105  82.164    98.012  83.032   
 99.025  84.028    99.730  84.730   ] scurve 
100.000  85.000 moveto 
110.000  85.000 moveto 
[ 110.000  85.000   110.105  83.852   110.155  83.615   110.215  83.378   
110.285  83.141   110.353  82.939   110.423  82.750   110.499  82.561   
110.582  82.372   110.671  82.184   110.768  81.996   110.871  81.809   
110.981  81.622   111.098  81.436   111.222  81.251   111.354  81.067   
111.493  80.884   111.640  80.702   111.774  80.544   111.903  80.400   
112.036  80.258   112.175  80.117   112.319  79.977   112.468  79.838   
112.623  79.702   112.784  79.567   112.949  79.433   113.121  79.302   
113.298  79.174   113.481  79.048   113.670  78.924   113.843  78.816   
114.003  78.722   114.166  78.630   114.334  78.540   114.505  78.452   
114.680  78.367   114.859  78.285   115.042  78.206   115.229  78.130   
115.420  78.057   115.615  77.987   115.814  77.921   116.016  77.859   
116.223  77.801   116.433  77.747   116.647  77.698   116.837  77.659   
117.014  77.626   117.193  77.597   117.374  77.571   117.557  77.549   
117.743  77.530   117.930  77.515   118.120  77.504   118.311  77.497   
118.505  77.494   118.700  77.495   118.897  77.501   119.095  77.511   
119.295  77.527   119.496  77.547   119.699  77.572   119.902  77.602   
120.107  77.637   120.312  77.678   120.517  77.724   120.724  77.776   
120.930  77.834   121.137  77.898   121.343  77.967   121.549  78.043   
121.755  78.125   121.959  78.213   122.163  78.307   122.365  78.408   
122.566  78.516   122.765  78.630   122.997  78.773   123.261  78.949   
123.590  79.193   124.138  79.662   124.918  80.522   125.527  81.437   
125.859  82.104   126.068  82.640   126.214  83.117   126.312  83.524   
126.383  83.919   126.427  84.260   126.453  84.580   126.465  84.902   
126.463  85.172   126.453  85.429   126.433  85.686   126.404  85.942   
126.366  86.196   126.319  86.449   126.264  86.699   126.199  86.947   
126.127  87.192   126.045  87.434   125.956  87.672   125.859  87.906   
125.753  88.136   125.641  88.362   125.521  88.583   125.394  88.799   
125.260  89.010   125.120  89.215   124.974  89.415   124.822  89.609   
124.665  89.797   124.502  89.979   124.334  90.154   124.162  90.324   
123.985  90.486   123.805  90.643   123.621  90.793   123.433  90.936   
123.242  91.073   123.049  91.203   122.853  91.326   122.654  91.443   
122.454  91.553   122.252  91.657   122.049  91.755   121.845  91.846   
121.639  91.930   121.433  92.009   121.227  92.081   121.020  92.147   
120.813  92.208   120.607  92.262   120.401  92.311   120.195  92.354   
119.990  92.392   119.786  92.424   119.583  92.452   119.381  92.474   
119.180  92.491   118.981  92.504   118.783  92.512   118.587  92.515   
118.392  92.514   118.200  92.509   118.009  92.499   117.821  92.486   
117.634  92.469   117.449  92.448   117.267  92.424   117.087  92.396   
116.909  92.364   116.734  92.330   116.561  92.293   116.390  92.252   
116.222  92.209   116.056  92.163   115.892  92.114   115.731  92.063   
115.573  92.010   115.417  91.954   115.264  91.896   115.113  91.836   
114.965  91.773   114.819  91.709   114.676  91.643   114.506  91.561   
114.335  91.474   114.159  91.380   113.955  91.264   113.744  91.137   
113.505  90.984   113.224  90.790   112.857  90.511   112.240  89.968   
111.469  89.114   110.890  88.265   110.564  87.640   110.360  87.148   
110.218  86.720   110.122  86.350   110.055  86.014   110.011  85.717   
109.983  85.421   109.971  85.178   109.968  84.941   109.975  84.704   
109.991  84.467   110.017  84.230   ] scurve 
123.000  85.000 moveto 
[ 123.000  85.000   122.807  86.124   122.755  86.254   122.705  86.364   
122.652  86.473   122.595  86.578   122.534  86.682   122.470  86.783   
122.402  86.881   122.331  86.976   122.256  87.068   122.178  87.158   
122.098  87.244   122.014  87.327   121.928  87.406   121.840  87.482   
121.749  87.555   121.656  87.624   121.561  87.690   121.464  87.751   
121.365  87.810   121.265  87.864   121.164  87.915   121.061  87.962   
120.957  88.005   120.852  88.045   120.747  88.080   120.640  88.112   
120.534  88.141   120.426  88.165   120.319  88.186   120.212  88.204   
120.104  88.218   119.997  88.228   119.890  88.234   119.784  88.238   
119.678  88.237   119.572  88.234   119.467  88.227   119.364  88.217   
119.261  88.204   119.159  88.187   119.058  88.168   118.958  88.146   
118.859  88.120   118.762  88.092   118.667  88.061   118.572  88.028   
118.480  87.991   118.388  87.953   118.299  87.911   118.211  87.868   
118.112  87.814   118.005  87.751   117.875  87.668   117.711  87.549   
117.419  87.296   116.912  86.659   116.734  86.322   116.627  86.060   
116.560  85.845   116.512  85.648   116.482  85.480   116.461  85.313   
116.450  85.177   116.445  85.041   116.446  84.904   116.453  84.768   
116.466  84.632   116.485  84.497   116.504  84.389   116.527  84.282   
116.553  84.175   116.584  84.069   116.618  83.964   116.655  83.860   
116.697  83.757   116.742  83.656   116.790  83.555   116.843  83.456   
116.899  83.359   116.958  83.263   117.021  83.169   117.088  83.076   
117.158  82.986   117.232  82.898   117.309  82.812   117.390  82.729   
117.474  82.647   117.561  82.569   117.651  82.493   117.745  82.421   
117.842  82.351   117.942  82.284   118.044  82.221   118.150  82.161   
118.239  82.115   118.327  82.072   118.417  82.031   118.509  81.993   
118.602  81.958   118.697  81.925   118.793  81.895   118.891  81.868   
118.990  81.843   119.090  81.822   119.192  81.804   119.294  81.788   
119.397  81.776   119.502  81.767   119.607  81.761   119.712  81.759   
119.819  81.760   119.926  81.764   120.033  81.772   120.140  81.783   
120.248  81.798   120.355  81.817   120.463  81.839   120.570  81.865   
120.677  81.895   120.783  81.928   120.888  81.965   120.993  82.006   
121.097  82.051   121.199  82.099   121.300  82.151   121.400  82.207   
121.520  82.280   121.659  82.374   121.852  82.523   122.322  83.006   
122.645  83.504   122.783  83.803   122.868  84.041   122.926  84.250   
122.965  84.436   122.990  84.601   123.006  84.753   123.014  84.900   
123.015  85.022   123.012  85.144   123.005  85.267   122.992  85.388   
122.975  85.509   122.954  85.629   122.928  85.748   122.897  85.866   
122.862  85.983   122.823  86.098   122.779  86.211   122.732  86.323   
122.680  86.432   122.624  86.539   122.565  86.644   122.502  86.746   
122.435  86.845   122.365  86.942   122.292  87.036   122.215  87.126   
122.135  87.214   122.053  87.298   121.968  87.380   121.880  87.457   
121.790  87.531   121.697  87.602   121.603  87.669   121.506  87.733   
121.408  87.793   121.308  87.849   121.207  87.901   121.105  87.950   
] scurve 
 90.000  85.000 moveto 
[  90.000  85.000    89.895  86.148    89.845  86.385    89.785  86.622   
 89.715  86.859    89.647  87.061    89.577  87.250    89.501  87.439   
 89.418  87.628    89.329  87.816    89.232  88.004    89.129  88.191   
 89.019  88.378    88.902  88.564    88.778  88.749    88.646  88.933   
 88.507  89.116    88.360  89.298    88.226  89.456    88.097  89.600   
 87.964  89.742    87.825  89.883    87.681  90.023    87.532  90.162   
 87.377  90.298    87.216  90.433    87.051  90.567    86.879  90.698   
 86.702  90.826    86.519  90.952    86.330  91.076    86.157  91.184   
 85.997  91.278    85.834  91.370    85.666  91.460    85.495  91.548   
 85.320  91.633    85.141  91.715    84.958  91.794    84.771  91.870   
 84.580  91.943    84.385  92.013    84.186  92.079    83.984  92.141   
 83.777  92.199    83.567  92.253    83.353  92.302    83.163  92.341   
 82.986  92.374    82.807  92.403    82.626  92.429    82.443  92.451   
 82.257  92.470    82.070  92.485    81.880  92.496    81.689  92.503   
 81.495  92.506    81.300  92.505    81.103  92.499    80.905  92.489   
 80.705  92.473    80.504  92.453    80.301  92.428    80.098  92.398   
 79.893  92.363    79.688  92.322    79.483  92.276    79.276  92.224   
 79.070  92.166    78.863  92.102    78.657  92.033    78.451  91.957   
 78.245  91.875    78.041  91.787    77.837  91.693    77.635  91.592   
 77.434  91.484    77.235  91.370    77.003  91.227    76.739  91.051   
 76.410  90.807    75.862  90.338    75.082  89.478    74.473  88.563   
 74.141  87.896    73.932  87.360    73.786  86.883    73.688  86.476   
 73.617  86.081    73.573  85.740    73.547  85.420    73.535  85.098   
 73.537  84.828    73.547  84.571    73.567  84.314    73.596  84.058   
 73.634  83.804    73.681  83.551    73.736  83.301    73.801  83.053   
 73.873  82.808    73.955  82.566    74.044  82.328    74.141  82.094   
 74.247  81.864    74.359  81.638    74.479  81.417    74.606  81.201   
 74.740  80.990    74.880  80.785    75.026  80.585    75.178  80.391   
 75.335  80.203    75.498  80.021    75.666  79.846    75.838  79.676   
 76.015  79.514    76.195  79.357    76.379  79.207    76.567  79.064   
 76.758  78.927    76.951  78.797    77.147  78.674    77.346  78.557   
 77.546  78.447    77.748  78.343    77.951  78.245    78.155  78.154   
 78.361  78.070    78.567  77.991    78.773  77.919    78.980  77.853   
 79.187  77.792    79.393  77.738    79.599  77.689    79.805  77.646   
 80.010  77.608    80.214  77.576    80.417  77.548    80.619  77.526   
 80.820  77.509    81.019  77.496    81.217  77.488    81.413  77.485   
 81.608  77.486    81.800  77.491    81.991  77.501    82.179  77.514   
 82.366  77.531    82.551  77.552    82.733  77.576    82.913  77.604   
 83.091  77.636    83.266  77.670    83.439  77.707    83.610  77.748   
 83.778  77.791    83.944  77.837    84.108  77.886    84.269  77.937   
 84.427  77.990    84.583  78.046    84.736  78.104    84.887  78.164   
 85.035  78.227    85.181  78.291    85.324  78.357    85.494  78.439   
 85.665  78.526    85.841  78.620    86.045  78.736    86.256  78.863   
 86.495  79.016    86.776  79.210    87.143  79.489    87.760  80.032   
 88.531  80.886    89.110  81.735    89.436  82.360    89.640  82.852   
 89.782  83.280    89.878  83.650    89.945  83.986    89.989  84.283   
 90.017  84.579    90.029  84.822    90.032  85.059    90.025  85.296   
 90.009  85.533    89.983  85.770   ] scurve 
 77.000  85.000 moveto 
[  77.000  85.000    77.193  83.876    77.245  83.746    77.295  83.636   
 77.348  83.527    77.405  83.422    77.466  83.318    77.530  83.217   
 77.598  83.119    77.669  83.024    77.744  82.932    77.822  82.842   
 77.902  82.756    77.986  82.673    78.072  82.594    78.160  82.518   
 78.251  82.445    78.344  82.376    78.439  82.310    78.536  82.249   
 78.635  82.190    78.735  82.136    78.836  82.085    78.939  82.038   
 79.043  81.995    79.148  81.955    79.253  81.920    79.360  81.888   
 79.466  81.859    79.574  81.835    79.681  81.814    79.788  81.796   
 79.896  81.782    80.003  81.772    80.110  81.766    80.216  81.762   
 80.322  81.763    80.428  81.766    80.533  81.773    80.636  81.783   
 80.739  81.796    80.841  81.813    80.942  81.832    81.042  81.854   
 81.141  81.880    81.238  81.908    81.333  81.939    81.428  81.972   
 81.520  82.009    81.612  82.047    81.701  82.089    81.789  82.132   
 81.888  82.186    81.995  82.249    82.125  82.332    82.289  82.451   
 82.581  82.704    83.088  83.341    83.266  83.678    83.373  83.940   
 83.440  84.155    83.488  84.352    83.518  84.520    83.539  84.687   
 83.550  84.823    83.555  84.959    83.554  85.096    83.547  85.232   
 83.534  85.368    83.515  85.503    83.496  85.611    83.473  85.718   
 83.447  85.825    83.416  85.931    83.382  86.036    83.345  86.140   
 83.303  86.243    83.258  86.344    83.210  86.445    83.157  86.544   
 83.101  86.641    83.042  86.737    82.979  86.831    82.912  86.924   
 82.842  87.014    82.768  87.102    82.691  87.188    82.610  87.271   
 82.526  87.353    82.439  87.431    82.349  87.507    82.255  87.579   
 82.158  87.649    82.058  87.716    81.956  87.779    81.850  87.839   
 81.761  87.885    81.673  87.928    81.583  87.969    81.491  88.007   
 81.398  88.042    81.303  88.075    81.207  88.105    81.109  88.132   
 81.010  88.157    80.910  88.178    80.808  88.196    80.706  88.212   
 80.603  88.224    80.498  88.233    80.393  88.239    80.288  88.241   
 80.181  88.240    80.074  88.236    79.967  88.228    79.860  88.217   
 79.752  88.202    79.645  88.183    79.537  88.161    79.430  88.135   
 79.323  88.105    79.217  88.072    79.112  88.035    79.007  87.994   
 78.903  87.949    78.801  87.901    78.700  87.849    78.600  87.793   
 78.480  87.720    78.341  87.626    78.148  87.477    77.678  86.994   
 77.355  86.496    77.217  86.197    77.132  85.959    77.074  85.750   
 77.035  85.564    77.010  85.399    76.994  85.247    76.986  85.100   
 76.985  84.978    76.988  84.856    76.995  84.733    77.008  84.612   
 77.025  84.491    77.046  84.371    77.072  84.252    77.103  84.134   
 77.138  84.017    77.177  83.902    77.221  83.789    77.268  83.677   
 77.320  83.568    77.376  83.461    77.435  83.356    77.498  83.254   
 77.565  83.155    77.635  83.058    77.708  82.964    77.785  82.874   
 77.865  82.786    77.947  82.702    78.032  82.620   ] scurve 
100.000 105.000 moveto 
[ 100.000 105.000    98.847 105.000    98.475 105.000    98.103 105.000   
 97.731 105.000    97.357 104.999    96.982 104.999    96.607 104.998   
 96.230 104.997    95.852 104.996    95.473 104.994    95.092 104.991   
 94.710 104.988    94.326 104.985    93.940 104.980    93.551 104.975   
 93.161 104.968    92.769 104.960    92.374 104.951    91.977 104.940   
 91.578 104.928    91.175 104.914    90.771 104.898    90.363 104.879   
 89.952 104.858    89.539 104.835    89.122 104.808    88.703 104.779   
 88.280 104.746    87.854 104.710    87.425 104.670    86.992 104.625   
 86.557 104.577    86.118 104.523    85.676 104.465    85.230 104.401   
 84.781 104.331    84.329 104.256    83.874 104.174    83.416 104.086   
 82.955 103.990    82.491 103.887    82.024 103.776    81.555 103.657   
 81.083 103.530    80.609 103.393    80.133 103.247    79.655 103.091   
 79.176 102.925    78.695 102.749    78.214 102.561    77.732 102.362   
 77.251 102.151    76.769 101.927    76.289 101.692    75.809 101.443   
 75.332 101.180    74.856 100.905    74.384 100.615    73.916 100.311   
 73.430  99.977    72.844  99.547    72.078  98.936    71.225  98.181   
 70.406  97.368    69.627  96.496    68.893  95.567    68.212  94.583   
 67.591  93.546    67.034  92.460    66.549  91.331    66.174  90.270   
 65.900  89.314    65.683  88.358    65.546  87.569    65.445  86.773   
 65.381  85.973    65.355  85.272    65.356  84.629    65.382  83.987   
 65.431  83.346    65.503  82.708    65.599  82.074    65.718  81.445   
 65.855  80.845    65.984  80.351    66.126  79.862    66.283  79.379   
 66.453  78.901    66.636  78.430    66.831  77.965    67.039  77.508   
 67.259  77.058    67.491  76.615    67.734  76.181    67.987  75.755   
 68.251  75.338    68.524  74.929    68.808  74.530    69.067  74.182   
 69.306  73.875    69.551  73.575    69.800  73.281    70.054  72.993   
 70.313  72.712    70.576  72.437    70.844  72.167    71.115  71.905   
 71.389  71.648    71.668  71.398    71.949  71.154    72.234  70.917   
 72.521  70.685    72.811  70.460    73.103  70.241    73.398  70.028   
 73.695  69.821    73.993  69.621    74.294  69.426    74.595  69.237   
 74.898  69.053    75.203  68.876    75.508  68.704    75.814  68.538   
 76.121  68.377    76.429  68.221    76.737  68.071    77.045  67.926   
 77.354  67.786    77.663  67.651    77.971  67.521    78.280  67.396   
 78.588  67.275    78.897  67.159    79.204  67.048    79.511  66.940   
 79.818  66.837    80.124  66.739    80.429  66.644    80.734  66.553   
 81.038  66.466    81.340  66.382    81.642  66.303    81.943  66.226   
 82.243  66.153    82.542  66.084    82.839  66.017    83.136  65.954   
 83.431  65.894    83.725  65.836    84.018  65.782    84.309  65.730   
 84.599  65.680    84.888  65.634    85.176  65.589    85.462  65.547   
 85.747  65.507    86.030  65.470    86.313  65.434    86.594  65.401   
 86.873  65.369    87.151  65.339    87.428  65.311    87.704  65.285   
 87.978  65.260    88.251  65.237    88.522  65.215    88.793  65.195   
 89.062  65.176    89.329  65.159    89.596  65.142    89.861  65.127   
 90.125  65.113    90.388  65.100    90.650  65.088    90.910  65.077   
 91.170  65.066    91.428  65.057    91.685  65.048    91.941  65.040   
 92.196  65.033    92.450  65.027    92.704  65.021    92.956  65.015   
 93.207  65.010    93.457  65.006    93.707  65.002    93.955  64.999   
 94.203  64.996    94.450  64.993   ] scurve 
[  94.450  64.993    94.697  64.990    94.942  64.988    95.187  64.987   
 95.431  64.985    95.675  64.984    95.918  64.983    96.160  64.982   
 96.402  64.981    96.644  64.980    96.884  64.980    97.125  64.979   
 97.365  64.979    97.605  64.979    97.844  64.978    98.083  64.978   
 98.322  64.978    98.561  64.978    98.799  64.978    99.037  64.978   
 99.275  64.978    99.513  64.978    99.751  64.978    99.989  64.978   
100.227  64.978   100.465  64.978   100.703  64.978   100.941  64.978   
101.179  64.978   101.418  64.978   101.656  64.978   101.895  64.978   
102.134  64.978   102.374  64.979   102.613  64.979   102.853  64.979   
103.094  64.980   103.335  64.980   103.576  64.981   103.818  64.981   
104.060  64.982   104.303  64.984   104.547  64.985   104.791  64.986   
105.036  64.988   105.281  64.990   105.527  64.993   105.774  64.995   
106.022  64.998   106.271  65.002   106.520  65.006   106.770  65.010   
107.022  65.015   107.274  65.020   107.527  65.026   107.781  65.032   
108.036  65.040   108.292  65.047   108.549  65.056   108.807  65.065   
109.066  65.076   109.327  65.087   109.588  65.099   109.851  65.112   
110.115  65.126   110.380  65.141   110.647  65.157   110.914  65.175   
111.183  65.194   111.453  65.214   111.725  65.235   111.998  65.258   
112.272  65.283   112.547  65.309   112.824  65.337   113.102  65.366   
113.381  65.398   113.662  65.431   113.944  65.467   114.228  65.504   
114.513  65.544   114.799  65.585   115.086  65.630   115.375  65.676   
115.665  65.725   115.957  65.777   116.249  65.831   116.543  65.889   
116.838  65.949   117.135  66.012   117.432  66.078   117.731  66.147   
118.031  66.220   118.332  66.296   118.633  66.375   118.936  66.459   
119.240  66.545   119.544  66.636   119.850  66.730   120.156  66.829   
120.462  66.932   120.770  67.039   121.077  67.150   121.385  67.266   
121.694  67.386   122.003  67.511   122.311  67.640   122.620  67.775   
122.929  67.915   123.238  68.059   123.546  68.209   123.853  68.364   
124.161  68.525   124.467  68.691   124.814  68.886   125.195  69.109   
125.573  69.341   126.026  69.633   126.502  69.958   127.077  70.375   
127.831  70.969   128.688  71.719   129.512  72.527   130.297  73.393   
131.037  74.317   131.725  75.297   132.355  76.330   132.919  77.412   
133.413  78.539   133.803  79.619   134.084  80.574   134.311  81.547   
134.453  82.337   134.558  83.133   134.626  83.934   134.656  84.665   
134.657  85.309   134.634  85.952   134.587  86.594   134.517  87.233   
134.423  87.868   134.306  88.499   134.169  89.112   134.041  89.607   
133.899  90.097   133.744  90.581   133.575  91.060   133.393  91.533   
133.198  91.999   132.991  92.458   132.772  92.909   132.541  93.353   
132.299  93.788   132.046  94.216   131.783  94.634   131.510  95.044   
131.227  95.445   130.967  95.794   130.729  96.102   130.484  96.403   
130.235  96.698   129.981  96.987   129.722  97.270   129.459  97.546   
129.192  97.816   128.921  98.080   128.646  98.337   128.368  98.589   
128.086  98.833   127.802  99.072   127.514  99.304   127.224  99.531   
126.932  99.751   126.637  99.964   126.340 100.172   126.041 100.374   
125.741 100.570   125.438 100.759   125.135 100.944   124.830 101.122   
124.525 101.294   124.218 101.462   123.911 101.623   123.603 101.779   
123.294 101.930   122.986 102.076   122.676 102.217   122.367 102.352   
122.058 102.483   121.749 102.609   ] scurve 
[ 121.749 102.609   121.440 102.730   121.132 102.847   120.823 102.959   
120.516 103.067   120.209 103.170   119.902 103.270   119.596 103.365   
119.291 103.457   118.987 103.544   118.684 103.628   118.381 103.708   
118.080 103.785   117.780 103.858   117.480 103.928   117.182 103.995   
116.885 104.059   116.590 104.120   116.295 104.177   116.002 104.232   
115.710 104.285   115.419 104.334   115.130 104.381   114.841 104.426   
114.555 104.468   114.269 104.508   113.985 104.546   113.702 104.582   
113.421 104.616   113.141 104.648   112.862 104.678   112.585 104.706   
112.309 104.732   112.034 104.757   111.760 104.781   111.488 104.803   
111.218 104.823   110.948 104.842   110.680 104.860   110.413 104.876   
110.147 104.892   109.882 104.906   109.619 104.919   109.357 104.932   
109.096 104.943   108.836 104.953   108.577 104.963   108.319 104.972   
108.063 104.980   107.807 104.987   107.553 104.994   107.299 105.000   
107.046 105.005   106.795 105.010   106.544 105.014   106.294 105.018   
106.045 105.022   105.796 105.025   105.549 105.028   105.302 105.030   
105.056 105.033   104.810 105.034   104.566 105.036   104.322 105.037   
104.078 105.039   103.835 105.040   103.593 105.040   103.351 105.041   
103.109 105.042   102.868 105.042   102.628 105.043   102.388 105.043   
102.148 105.043   101.908 105.043   101.669 105.043   101.430 105.043   
101.191 105.044   100.952 105.044   100.713 105.044   100.475 105.044   
100.237 105.044    99.998 105.044    99.760 105.044    99.521 105.044   
 99.283 105.044    99.044 105.044    98.806 105.044    98.567 105.043   
 98.328 105.043    98.088 105.043    97.849 105.043    97.609 105.043   
 97.368 105.043    97.128 105.042    96.887 105.042    96.645 105.041   
 96.403 105.040    96.161 105.040    95.918 105.039    95.675 105.037   
 95.431 105.036    95.186 105.034    94.940 105.032    94.694 105.030   
 94.448 105.028    94.200 105.025   ] scurve 
100.000  94.000 moveto 
[ 100.000  94.000    98.852  94.048    98.612  94.070    98.371  94.096   
 98.128  94.127    97.883  94.161    97.637  94.199    97.389  94.241   
 97.139  94.287    96.886  94.337    96.630  94.391    96.371  94.448   
 96.110  94.509    95.844  94.574    95.575  94.642    95.302  94.713   
 95.051  94.781    94.826  94.843    94.599  94.907    94.368  94.972   
 94.134  95.039    93.897  95.108    93.656  95.179    93.411  95.251   
 93.163  95.324    92.910  95.399    92.653  95.475    92.392  95.551   
 92.127  95.629    91.856  95.707    91.581  95.785    91.347  95.851   
 91.120  95.914    90.889  95.977    90.655  96.040    90.417  96.102   
 90.176  96.164    89.931  96.226    89.681  96.286    89.429  96.346   
 89.172  96.405    88.911  96.462    88.645  96.518    88.376  96.573   
 88.102  96.625    87.824  96.676    87.557  96.721    87.328  96.758   
 87.095  96.793    86.860  96.827    86.622  96.858    86.381  96.887   
 86.137  96.914    85.890  96.938    85.640  96.960    85.387  96.979   
 85.131  96.995    84.872  97.009    84.610  97.018    84.345  97.025   
 84.077  97.027    83.806  97.026    83.532  97.021    83.256  97.012   
 82.976  96.998    82.694  96.980    82.409  96.956    82.122  96.928   
 81.832  96.894    81.540  96.855    81.246  96.810    80.949  96.759   
 80.651  96.702    80.351  96.638    80.049  96.568    79.746  96.490   
 79.441  96.406    79.136  96.314    78.829  96.214    78.522  96.107   
 78.215  95.992    77.908  95.868    77.601  95.736    77.295  95.595   
 76.990  95.445    76.686  95.287    76.384  95.119    76.084  94.942   
 75.780  94.751    75.412  94.504    74.976  94.186    74.349  93.675   
 73.533  92.897    72.779  92.030    72.102  91.077    71.517  90.047   
 71.122  89.173    70.859  88.447    70.666  87.783    70.528  87.175   
 70.428  86.593    70.367  86.096    70.329  85.596    70.313  85.104   
 70.316  84.703    70.334  84.302    70.366  83.902    70.412  83.504   
 70.473  83.109    70.547  82.716    70.635  82.327    70.737  81.942   
 70.852  81.562    70.980  81.187    71.120  80.818    71.273  80.455   
 71.425  80.125    71.565  79.844    71.711  79.569    71.864  79.298   
 72.024  79.033    72.191  78.773    72.363  78.518    72.541  78.269   
 72.725  78.025    72.915  77.788    73.109  77.556    73.308  77.331   
 73.512  77.111    73.720  76.898    73.933  76.691    74.149  76.490   
 74.368  76.295    74.591  76.107    74.817  75.924    75.046  75.749   
 75.278  75.579    75.511  75.415    75.747  75.258    75.985  75.107   
 76.225  74.962    76.465  74.823    76.708  74.690    76.951  74.563   
 77.195  74.441    77.440  74.325    77.685  74.215    77.931  74.111   
 78.177  74.012    78.423  73.918    78.669  73.829    78.914  73.745   
 79.160  73.667    79.404  73.593    79.648  73.524    79.892  73.460   
 80.134  73.400    80.376  73.344    80.616  73.293    80.856  73.246   
 81.094  73.203    81.331  73.164    81.566  73.129    81.800  73.097   
 82.033  73.069    82.264  73.045    82.493  73.024    82.721  73.006   
 82.947  72.991    83.171  72.979    83.394  72.970    83.615  72.964   
 83.834  72.960    84.051  72.959    84.266  72.961    84.479  72.964   
 84.691  72.970    84.900  72.979    85.108  72.989    85.313  73.001   
 85.517  73.016    85.719  73.032    85.919  73.050    86.117  73.069   
 86.313  73.090    86.507  73.113    86.699  73.137    86.889  73.162   
 87.078  73.188    87.264  73.216   ] scurve 
[  87.264  73.216    87.449  73.245    87.631  73.275    87.812  73.306   
 87.991  73.338    88.169  73.371    88.344  73.404    88.518  73.439   
 88.690  73.474    88.860  73.509    89.028  73.546    89.195  73.582   
 89.360  73.620    89.524  73.658    89.686  73.696    89.846  73.735   
 90.004  73.773    90.162  73.813    90.317  73.852    90.471  73.892   
 90.624  73.932    90.775  73.972    90.924  74.012    91.072  74.052   
 91.219  74.092    91.364  74.132    91.508  74.173    91.651  74.213   
 91.792  74.253    91.932  74.293    92.071  74.333    92.209  74.373   
 92.345  74.412    92.480  74.451    92.614  74.491    92.747  74.529   
 92.879  74.568    93.009  74.606    93.139  74.644    93.267  74.682   
 93.394  74.719    93.544  74.763    93.700  74.809    93.855  74.854   
 94.008  74.898    94.160  74.942    94.310  74.985    94.459  75.028   
 94.607  75.069    94.754  75.110    94.899  75.150    95.043  75.189   
 95.186  75.228    95.328  75.265    95.468  75.302    95.608  75.338   
 95.746  75.373    95.884  75.407    96.021  75.440    96.156  75.473   
 96.291  75.504    96.425  75.534    96.558  75.564    96.690  75.592   
 96.822  75.620    96.953  75.646    97.083  75.671    97.212  75.696   
 97.341  75.719    97.469  75.742    97.597  75.763    97.724  75.783   
 97.851  75.803    97.977  75.821    98.102  75.838    98.227  75.854   
 98.352  75.869    98.477  75.883    98.601  75.896    98.725  75.908   
 98.848  75.919    98.971  75.929    99.094  75.937    99.217  75.945   
 99.340  75.951    99.462  75.956    99.584  75.961    99.707  75.964   
 99.829  75.966    99.951  75.967   100.073  75.967   100.195  75.966   
100.317  75.963   100.440  75.960   100.562  75.955   100.685  75.950   
100.807  75.943   100.930  75.935   101.053  75.927   101.176  75.917   
101.300  75.906   101.424  75.894   101.548  75.881   101.672  75.866   
101.797  75.851   101.922  75.835   102.048  75.817   102.174  75.799   
102.301  75.779   102.428  75.759   102.556  75.737   102.684  75.715   
102.813  75.691   102.943  75.666   103.073  75.641   103.204  75.614   
103.336  75.587   103.468  75.558   103.601  75.528   103.735  75.498   
103.870  75.466   104.006  75.434   104.143  75.400   104.281  75.366   
104.419  75.331   104.559  75.295   104.700  75.258   104.842  75.220   
104.985  75.182   105.129  75.142   105.275  75.102   105.421  75.061   
105.569  75.019   105.719  74.977   105.870  74.933   106.022  74.890   
106.175  74.845   106.330  74.800   106.487  74.754   106.645  74.708   
106.805  74.661   106.966  74.613   107.129  74.565   107.294  74.517   
107.460  74.468   107.629  74.419   107.799  74.370   107.971  74.320   
108.145  74.270   108.322  74.220   108.500  74.170   108.680  74.120   
108.862  74.069   109.047  74.019   109.234  73.969   109.423  73.919   
109.614  73.869   109.808  73.820   110.004  73.771   110.203  73.722   
110.404  73.674   110.608  73.627   110.814  73.580   111.023  73.534   
111.234  73.489   111.448  73.445   111.665  73.402   111.885  73.360   
112.107  73.319   112.333  73.280   112.561  73.243   112.792  73.207   
113.026  73.173   113.263  73.141   113.503  73.111   113.746  73.083   
113.992  73.057   114.241  73.034   114.493  73.014   114.748  72.997   
115.007  72.982   115.268  72.971   115.532  72.963   115.799  72.959   
116.069  72.958   116.342  72.962   116.618  72.969   116.896  72.981   
117.178  72.998   117.462  73.019   ] scurve 
[ 117.462  73.019   117.748  73.045   118.037  73.077   118.329  73.114   
118.623  73.156   118.919  73.205   119.217  73.259   119.516  73.320   
119.818  73.388   120.121  73.462   120.426  73.544   120.731  73.633   
121.038  73.729   121.345  73.833   121.652  73.945   121.960  74.065   
122.267  74.194   122.574  74.331   122.880  74.477   123.185  74.632   
123.488  74.796   123.790  74.970   124.089  75.152   124.439  75.381   
124.837  75.661   125.344  76.053   126.122  76.741   126.906  77.571   
127.619  78.488   128.247  79.487   128.724  80.439   129.043  81.236   
129.275  81.962   129.427  82.566   129.547  83.180   129.620  83.699   
129.668  84.198   129.693  84.700   129.696  85.138   129.684  85.540   
129.658  85.941   129.617  86.340   129.562  86.737   129.493  87.132   
129.410  87.523   129.314  87.910   129.204  88.293   129.081  88.670   
128.945  89.042   128.797  89.408   128.637  89.768   128.500  90.050   
128.356  90.328   128.205  90.601   128.048  90.869   127.884  91.132   
127.713  91.389   127.537  91.640   127.355  91.886   127.168  92.127   
126.975  92.361   126.777  92.589   126.575  92.812   126.368  93.028   
126.157  93.238   125.942  93.441   125.723  93.639   125.501  93.830   
125.276  94.015   125.048  94.193   124.817  94.366   124.584  94.532   
124.349  94.692   124.111  94.845   123.872  94.993   123.631  95.135   
123.389  95.270   123.146  95.400   122.902  95.524   122.657  95.642   
122.411  95.754   122.165  95.861   121.919  95.963   121.673  96.059   
121.427  96.150   121.181  96.235   120.935  96.316   120.690  96.392   
120.445  96.463   120.201  96.529   119.958  96.591   119.715  96.648   
119.474  96.701   119.234  96.749   118.995  96.794   118.757  96.835   
118.521  96.871   118.286  96.904   118.053  96.934   117.821  96.960   
117.590  96.982   117.362  97.002   117.135  97.018   116.909  97.031   
116.686  97.041   116.464  97.048   116.244  97.053   116.026  97.055   
115.810  97.055   115.596  97.052   115.383  97.047   115.173  97.039   
114.964  97.030   114.757  97.018   114.553  97.005   114.350  96.989   
114.149  96.972   113.950  96.953   113.753  96.933   113.558  96.911   
113.365  96.888   113.173  96.863   112.984  96.837   112.797  96.810   
112.611  96.781   112.427  96.752   112.245  96.721   112.065  96.690   
111.887  96.657   111.711  96.624   111.536  96.590   111.363  96.555   
111.192  96.520   111.023  96.484   110.855  96.447   110.689  96.410   
110.525  96.373   110.362  96.335   110.201  96.296   110.042  96.257   
109.884  96.218   109.727  96.179   109.573  96.139   109.419  96.100   
109.267  96.060   109.117  96.020   108.968  95.980   108.821  95.940   
108.674  95.900   108.530  95.859   108.386  95.819   108.244  95.779   
108.103  95.739   107.964  95.700   107.826  95.660   107.689  95.620   
107.553  95.581   107.418  95.542   107.285  95.503   107.152  95.465   
107.021  95.426   106.891  95.388   106.762  95.351   106.634  95.313   
106.507  95.276   106.361  95.234   106.205  95.188   106.051  95.144   
105.898  95.100   105.746  95.057   105.596  95.014   105.447  94.972   
105.300  94.931   105.154  94.891   105.009  94.851   104.865  94.813   
104.722  94.775   104.581  94.738   104.440  94.702   104.301  94.667   
104.162  94.632   104.025  94.599   103.888  94.566   103.753  94.535   
103.618  94.504   103.484  94.474   103.351  94.446   103.218  94.418   
103.087  94.391   102.956  94.366   ] scurve 
[ 102.956  94.366   102.826  94.341   102.696  94.317   102.567  94.295   
102.439  94.273   102.311  94.252   102.184  94.233   102.057  94.214   
101.931  94.197   101.805  94.181   101.679  94.165   101.554  94.151   
101.430  94.138   101.305  94.126   101.181  94.115   101.057  94.105   
100.934  94.096   100.810  94.088   100.687  94.082   100.564  94.076   
100.441  94.072   100.318  94.068   100.195  94.066   100.073  94.065   
 99.950  94.065   ] scurve
100.000 100.000 moveto 
[ 100.000 100.000    98.847 100.012    98.542 100.020    98.243 100.028   
 97.944 100.039    97.643 100.051    97.342 100.064    97.039 100.078   
 96.735 100.095    96.429 100.112    96.122 100.130    95.812 100.150   
 95.501 100.171    95.188 100.193    94.872 100.216    94.555 100.239   
 94.234 100.263    93.911 100.288    93.585 100.313    93.256 100.338   
 92.924 100.364    92.595 100.389    92.325 100.409    92.052 100.429   
 91.777 100.449    91.500 100.468    91.220 100.487    90.937 100.505   
 90.653 100.523    90.365 100.539    90.075 100.555    89.782 100.569   
 89.486 100.582    89.188 100.594    88.886 100.604    88.582 100.613   
 88.274 100.619    87.964 100.624    87.650 100.626    87.333 100.626   
 87.013 100.623    86.690 100.618    86.364 100.609    86.035 100.597   
 85.702 100.582    85.367 100.563    85.028 100.540    84.685 100.513   
 84.340 100.482    83.992 100.446    83.640 100.405    83.286 100.359   
 82.929 100.307    82.569 100.250    82.206 100.187    81.840 100.117   
 81.472 100.041    81.102  99.958    80.729  99.868    80.355  99.771   
 79.979  99.665    79.601  99.552    79.221  99.430    78.841  99.299   
 78.460  99.159    78.078  99.011    77.696  98.852    77.314  98.684   
 76.932  98.505    76.552  98.316    76.172  98.117    75.795  97.906   
 75.419  97.684    75.047  97.451    74.643  97.184    74.177  96.853   
 73.566  96.382    72.749  95.674    71.942  94.869    71.184  93.993   
 70.486  93.048    69.857  92.039    69.306  90.969    68.895  89.993   
 68.586  89.085    68.380  88.327    68.214  87.557    68.107  86.890   
 68.036  86.261    67.993  85.629    67.978  85.042    67.986  84.535   
 68.011  84.028    68.055  83.524    68.117  83.021    68.197  82.522   
 68.294  82.027    68.409  81.536    68.541  81.051    68.689  80.572   
 68.839  80.142    68.981  79.769    69.133  79.402    69.294  79.039   
 69.465  78.682    69.645  78.331    69.833  77.986    70.031  77.648   
 70.236  77.316    70.449  76.991    70.669  76.672    70.897  76.361   
 71.131  76.057    71.372  75.761    71.619  75.472    71.872  75.190   
 72.130  74.917    72.394  74.650    72.662  74.392    72.935  74.141   
 73.212  73.898    73.493  73.663    73.777  73.435    74.064  73.215   
 74.355  73.003    74.648  72.798    74.943  72.601    75.241  72.411   
 75.541  72.228    75.842  72.053    76.144  71.884    76.448  71.723   
 76.752  71.569    77.057  71.421    77.363  71.280    77.669  71.145   
 77.975  71.017    78.281  70.895    78.586  70.779    78.891  70.669   
 79.196  70.564    79.500  70.465    79.803  70.372    80.105  70.284   
 80.407  70.201    80.707  70.123    81.005  70.050    81.303  69.981   
 81.598  69.917    81.893  69.857    82.185  69.802    82.476  69.750   
 82.766  69.703    83.053  69.659    83.339  69.619    83.623  69.582   
 83.905  69.548    84.185  69.518    84.463  69.491    84.739  69.466   
 85.013  69.445    85.285  69.426    85.555  69.410    85.822  69.396   
 86.088  69.384    86.352  69.375    86.614  69.367    86.874  69.362   
 87.131  69.359    87.387  69.357    87.641  69.357    87.892  69.358   
 88.142  69.361    88.390  69.365    88.636  69.371    88.879  69.377   
 89.121  69.385    89.361  69.394    89.600  69.404    89.836  69.415   
 90.071  69.426    90.304  69.438    90.535  69.451    90.764  69.465   
 90.992  69.479    91.218  69.493    91.443  69.508    91.666  69.523   
 91.888  69.539    92.108  69.555   ] scurve 
[  92.108  69.555    92.326  69.570    92.543  69.587    92.759  69.603   
 92.973  69.619    93.186  69.635    93.398  69.651    93.609  69.668   
 93.818  69.684    94.026  69.699    94.233  69.715    94.439  69.731   
 94.644  69.746    94.848  69.761    95.051  69.775    95.252  69.790   
 95.453  69.803    95.654  69.817    95.853  69.830    96.051  69.843   
 96.249  69.855    96.446  69.866    96.643  69.877    96.838  69.888   
 97.033  69.898    97.228  69.908    97.422  69.917    97.615  69.925   
 97.808  69.933    98.001  69.940    98.193  69.947    98.385  69.952   
 98.577  69.958    98.768  69.962    98.959  69.966    99.150  69.970   
 99.341  69.972    99.531  69.974    99.722  69.976    99.912  69.976   
100.102  69.976   100.293  69.976   100.483  69.974   100.674  69.972   
100.865  69.969   101.055  69.966   101.247  69.962   101.438  69.957   
101.629  69.952   101.821  69.946   102.014  69.939   102.206  69.932   
102.399  69.924   102.593  69.916   102.787  69.907   102.981  69.897   
103.177  69.887   103.372  69.877   103.569  69.865   103.766  69.854   
103.964  69.842   104.162  69.829   104.362  69.816   104.562  69.802   
104.763  69.788   104.965  69.774   105.168  69.760   105.372  69.745   
105.577  69.729   105.782  69.714   105.990  69.698   106.198  69.682   
106.407  69.666   106.618  69.650   106.830  69.634   107.043  69.618   
107.257  69.602   107.473  69.585   107.690  69.569   107.909  69.553   
108.129  69.538   108.351  69.522   108.574  69.507   108.798  69.492   
109.025  69.478   109.253  69.464   109.482  69.450   109.713  69.437   
109.946  69.425   110.181  69.414   110.418  69.403   110.656  69.393   
110.896  69.385   111.139  69.377   111.383  69.370   111.629  69.365   
111.876  69.361   112.126  69.358   112.378  69.356   112.632  69.357   
112.888  69.359   113.146  69.362   113.405  69.368   113.667  69.375   
113.931  69.385   114.197  69.397   114.465  69.411   114.736  69.427   
115.008  69.446   115.282  69.468   115.558  69.493   115.836  69.520   
116.116  69.551   116.398  69.584   116.682  69.621   116.968  69.662   
117.256  69.706   117.545  69.754   117.837  69.806   118.130  69.862   
118.424  69.922   118.720  69.986   119.018  70.055   119.316  70.129   
119.616  70.207   119.918  70.291   120.220  70.379   120.523  70.473   
120.828  70.572   121.132  70.677   121.438  70.788   121.743  70.904   
122.050  71.027   122.356  71.156   122.662  71.291   122.967  71.433   
123.272  71.581   123.577  71.736   123.881  71.898   124.183  72.067   
124.491  72.247   124.865  72.477   125.244  72.725   125.703  73.044   
126.269  73.471   127.048  74.125   127.868  74.914   128.640  75.773   
129.354  76.703   130.002  77.698   130.575  78.755   131.030  79.783   
131.359  80.703   131.587  81.493   131.762  82.261   131.884  82.970   
131.962  83.599   132.011  84.231   132.032  84.854   132.028  85.362   
132.006  85.869   131.966  86.375   131.908  86.879   131.831  87.379   
131.737  87.876   131.626  88.368   131.497  88.855   131.352  89.336   
131.199  89.785   131.059  90.160   130.909  90.529   130.749  90.893   
130.580  91.252   130.402  91.605   130.214  91.951   130.019  92.291   
129.815  92.625   129.603  92.952   129.384  93.272   129.157  93.585   
128.924  93.891   128.684  94.189   128.438  94.480   128.186  94.764   
127.928  95.039   127.665  95.308   127.397  95.568   127.125  95.821   
126.849  96.066   126.568  96.303   ] scurve 
[ 126.568  96.303   126.284  96.532   125.997  96.754   125.707  96.968   
125.413  97.175   125.118  97.374   124.820  97.565   124.521  97.750   
124.220  97.927   123.917  98.097   123.613  98.260   123.308  98.416   
123.003  98.565   122.697  98.707   122.391  98.843   122.084  98.973   
121.778  99.097   121.472  99.214   121.166  99.325   120.861  99.431   
120.556  99.531   120.253  99.626   119.950  99.715   119.648  99.799   
119.348  99.878   119.048  99.952   118.750 100.022   118.454 100.087   
118.159 100.147   117.865 100.204   117.573 100.256   117.283 100.305   
116.995 100.349   116.709 100.390   116.424 100.428   116.142 100.462   
115.861 100.493   115.582 100.521   115.305 100.546   115.031 100.568   
114.758 100.587   114.487 100.604   114.218 100.619   113.952 100.631   
113.687 100.641   113.425 100.648   113.164 100.654   112.906 100.658   
112.649 100.660   112.395 100.661   112.143 100.660   111.892 100.657   
111.644 100.653   111.397 100.648   111.153 100.642   110.910 100.634   
110.669 100.625   110.430 100.616   110.193 100.605   109.958 100.594   
109.724 100.582   109.492 100.569   109.262 100.556   109.034 100.542   
108.807 100.528   108.582 100.513   108.358 100.498   108.136 100.483   
107.915 100.467   107.696 100.451   107.478 100.435   107.262 100.419   
107.047 100.403   106.833 100.387   106.621 100.371   106.410 100.355   
106.200 100.339   105.991 100.323   105.784 100.307   105.577 100.292   
105.372 100.277   105.167 100.262   104.964 100.247   104.761 100.233   
104.560 100.219   104.359 100.206   104.159 100.193   103.960 100.180   
103.762 100.168   103.564 100.157   103.368 100.145   103.171 100.135   
102.976 100.125   102.781 100.115   102.586 100.106   102.392 100.098   
102.199 100.090   102.005 100.083   101.813 100.076   101.620 100.071   
101.428 100.065   101.236 100.061   101.045 100.057   100.854 100.053   
100.662 100.051   100.471 100.049   100.280 100.047   100.089 100.047   
 99.898 100.047    99.708 100.047    99.517 100.049    99.325 100.051   
 99.134 100.054    98.943 100.057    98.751 100.061    98.560 100.066   
 98.367 100.071    98.175 100.077    97.982 100.083    97.789 100.091   
 97.596 100.098    97.401 100.107    97.207 100.116    97.012 100.125   
 96.816 100.135    96.620 100.146    96.423 100.157    96.225 100.169   
 96.027 100.181    95.828 100.194    95.628 100.207    95.427 100.220   
 95.226 100.234    95.023 100.248    94.820 100.263    94.615 100.278   
 94.410 100.293    94.203 100.308    93.996 100.324    93.787 100.340   
 93.577 100.356    93.366 100.372    93.153 100.388    92.940 100.404   
 92.724 100.420    92.508 100.436    92.290 100.452   ] scurve 
100.000 111.000 moveto 
[ 100.000 111.000    98.830 110.993    98.366 110.987    97.902 110.978   
 97.437 110.967    96.972 110.954    96.506 110.939    96.039 110.921   
 95.572 110.901    95.104 110.879    94.634 110.854    94.164 110.826   
 93.692 110.795    93.219 110.762    92.745 110.726    92.269 110.687   
 91.792 110.644    91.313 110.598    90.833 110.549    90.351 110.496   
 89.867 110.439    89.381 110.378    88.894 110.314    88.405 110.244   
 87.914 110.171    87.421 110.092    86.926 110.009    86.430 109.921   
 85.931 109.827    85.431 109.727    84.929 109.622    84.424 109.511   
 83.919 109.393    83.411 109.269    82.902 109.138    82.391 109.000   
 81.879 108.854    81.366 108.701    80.851 108.540    80.335 108.371   
 79.819 108.193    79.302 108.006    78.784 107.810    78.266 107.605   
 77.748 107.390    77.230 107.165    76.712 106.929    76.195 106.684   
 75.680 106.427    75.165 106.159    74.653 105.880    74.142 105.589   
 73.634 105.286    73.129 104.971    72.628 104.644    72.051 104.249   
 71.408 103.783    70.600 103.156    69.675 102.375    68.777 101.542   
 67.911 100.659    67.080  99.725    66.291  98.740    65.547  97.706   
 64.853  96.624    64.215  95.498    63.636  94.329    63.122  93.122   
 62.676  91.880    62.301  90.609    62.056  89.573    61.860  88.525   
 61.714  87.466    61.626  86.497    61.584  85.641    61.575  84.784   
 61.600  83.927    61.658  83.073    61.748  82.222    61.854  81.483   
 61.973  80.811    62.112  80.143    62.272  79.481    62.451  78.825   
 62.649  78.176    62.866  77.536    63.102  76.903    63.356  76.280   
 63.612  75.699    63.840  75.215    64.079  74.737    64.328  74.266   
 64.587  73.803    64.856  73.347    65.134  72.898    65.421  72.457   
 65.716  72.025    66.020  71.600    66.332  71.183    66.651  70.774   
 66.978  70.374    67.312  69.983    67.652  69.600    67.999  69.225   
 68.351  68.859    68.709  68.502    69.073  68.153    69.442  67.813   
 69.815  67.481    70.193  67.158    70.512  66.894    70.820  66.648   
 71.130  66.406    71.442  66.171    71.757  65.940    72.073  65.715   
 72.391  65.495    72.711  65.280    73.033  65.070    73.355  64.865   
 73.679  64.666    74.005  64.471    74.331  64.282    74.659  64.097   
 74.987  63.916    75.316  63.741    75.646  63.570    75.976  63.404   
 76.307  63.242    76.638  63.085    76.969  62.932    77.301  62.784   
 77.633  62.639    77.965  62.499    78.297  62.363    78.629  62.230   
 78.961  62.102    79.293  61.977    79.624  61.856    79.956  61.739   
 80.287  61.626    80.618  61.516    80.948  61.409    81.278  61.306   
 81.607  61.206    81.936  61.109    82.264  61.015    82.592  60.925   
 82.919  60.837    83.246  60.752    83.571  60.671    83.897  60.592   
 84.221  60.515    84.545  60.441    84.868  60.370    85.190  60.302   
 85.512  60.235    85.833  60.172    86.153  60.110    86.472  60.051   
 86.791  59.994    87.108  59.939    87.425  59.886    87.742  59.835   
 88.057  59.786    88.372  59.739    88.686  59.694    88.999  59.651   
 89.311  59.609    89.623  59.570    89.934  59.531    90.244  59.495   
 90.554  59.460    90.862  59.426    91.171  59.395    91.478  59.364   
 91.785  59.335    92.091  59.307    92.397  59.281    92.702  59.256   
 93.006  59.232    93.310  59.209    93.613  59.188    93.916  59.168   
 94.218  59.149    94.520  59.131    94.821  59.114    95.122  59.098   
 95.423  59.083    95.723  59.070   ] scurve 
[  95.723  59.070    96.022  59.057    96.322  59.045    96.621  59.035   
 96.920  59.025    97.218  59.016    97.517  59.008    97.815  59.001   
 98.112  58.995    98.410  58.990    98.708  58.986    99.005  58.982   
 99.303  58.980    99.600  58.978    99.897  58.978   100.194  58.978   
100.492  58.979   100.789  58.981   101.086  58.983   101.384  58.987   
101.681  58.992   101.979  58.997   102.277  59.003   102.575  59.011   
102.874  59.019   103.172  59.028   103.471  59.038   103.770  59.049   
104.070  59.061   104.370  59.074   104.670  59.088   104.970  59.103   
105.272  59.119   105.573  59.136   105.875  59.154   106.177  59.174   
106.480  59.194   106.784  59.216   107.088  59.239   107.392  59.263   
107.698  59.289   108.003  59.316   108.310  59.344   108.617  59.373   
108.925  59.404   109.233  59.437   109.542  59.471   109.852  59.506   
110.162  59.543   110.473  59.582   110.785  59.622   111.098  59.664   
111.411  59.708   111.726  59.754   112.041  59.801   112.356  59.851   
112.673  59.902   112.990  59.956   113.308  60.011   113.627  60.069   
113.946  60.129   114.267  60.191   114.588  60.256   114.910  60.323   
115.232  60.392   115.556  60.464   115.880  60.539   116.205  60.616   
116.530  60.696   116.856  60.779   117.183  60.864   117.510  60.953   
117.838  61.044   118.167  61.139   118.496  61.237   118.826  61.338   
119.156  61.442   119.486  61.550   119.817  61.661   120.148  61.776   
120.480  61.894   120.812  62.016   121.143  62.142   121.476  62.272   
121.808  62.405   122.140  62.543   122.472  62.685   122.804  62.830   
123.136  62.980   123.467  63.135   123.799  63.293   124.129  63.457   
124.460  63.624   124.789  63.797   125.163  63.998   125.573  64.227   
125.981  64.463   126.388  64.707   126.884  65.017   127.387  65.345   
127.964  65.741   128.622  66.218   129.448  66.861   130.373  67.646   
131.271  68.481   132.137  69.368   132.967  70.306   133.756  71.295   
134.498  72.333   135.191  73.418   135.827  74.548   136.403  75.721   
136.915  76.931   137.359  78.176   137.730  79.451   137.973  80.489   
138.166  81.540   138.308  82.601   138.392  83.552   138.432  84.409   
138.439  85.268   138.413  86.125   138.353  86.981   138.260  87.833   
138.154  88.557   138.034  89.231   137.894  89.899   137.733  90.562   
137.553  91.218   137.353  91.867   137.134  92.508   136.897  93.141   
136.642  93.765   136.390  94.335   136.160  94.820   135.921  95.297   
135.671  95.768   135.411  96.232   135.141  96.689   134.862  97.137   
134.574  97.578   134.278  98.012   133.973  98.437   133.660  98.853   
133.340  99.262   133.012  99.662   132.678 100.054   132.337 100.437   
131.989 100.812   131.636 101.178   131.276 101.535   130.912 101.884   
130.543 102.224   130.168 102.555   129.790 102.878   129.484 103.131   
129.175 103.377   128.865 103.619   128.552 103.855   128.237 104.086   
127.920 104.311   127.601 104.531   127.281 104.746   126.959 104.956   
126.635 105.161   126.311 105.360   125.985 105.555   125.658 105.745   
125.330 105.930   125.001 106.110   124.672 106.285   124.342 106.456   
124.011 106.623   123.679 106.784   123.348 106.942   123.016 107.095   
122.683 107.243   122.351 107.388   122.018 107.528   121.686 107.664   
121.353 107.797   121.021 107.925   120.689 108.050   120.356 108.170   
120.025 108.287   119.693 108.401   119.362 108.511   119.031 108.618   
118.701 108.721   118.371 108.821   ] scurve 
[ 118.371 108.821   118.042 108.918   117.713 109.011   117.385 109.102   
117.057 109.189   116.730 109.274   116.404 109.356   116.078 109.435   
115.753 109.511   115.429 109.585   115.105 109.656   114.782 109.725   
114.460 109.791   114.139 109.855   113.818 109.916   113.499 109.975   
113.180 110.032   112.862 110.087   112.544 110.140   112.227 110.191   
111.911 110.239   111.596 110.286   111.282 110.331   110.968 110.375   
110.655 110.416   110.343 110.456   110.032 110.494   109.721 110.530   
109.411 110.565   109.102 110.599   108.793 110.631   108.485 110.661   
108.178 110.690   107.871 110.718   107.565 110.744   107.260 110.769   
106.955 110.793   106.651 110.815   106.347 110.837   106.044 110.857   
105.741 110.876   105.438 110.894   105.137 110.910   104.835 110.926   
104.534 110.941   104.234 110.954   103.933 110.967   103.633 110.978   
103.334 110.989   103.035 110.999   102.736 111.007   102.437 111.015   
102.138 111.022   101.840 111.028   101.541 111.033   101.243 111.037   
100.945 111.040   100.647 111.043   100.350 111.044   100.052 111.045   
 99.754 111.044    99.456 111.043    99.158 111.041    98.860 111.038   
 98.562 111.034    98.264 111.030    97.965 111.024    97.667 111.018   
 97.368 111.010    97.069 111.002    96.770 110.992    96.470 110.982   
 96.171 110.971    95.871 110.959    95.570 110.945    95.269 110.931   
 94.968 110.916    94.666 110.899    94.364 110.882    94.061 110.863   
 93.758 110.844    93.455 110.823    93.151 110.801    92.846 110.777   
] scurve 
currentdash currentlinewidth grestore setlinewidth setdash gsave 
 55.000  55.000 moveto 
 90.000  60.000 rect  clip  newpath 
 63.654 115.349 moveto 
[  63.654 115.349    64.335 114.530    65.013 113.708    65.687 112.882   
 66.357 112.052    67.024 111.219    67.686 110.381    68.343 109.538   
 68.996 108.691    69.643 107.838    70.284 106.979    70.918 106.113   
 71.545 105.240    72.165 104.360    72.776 103.471    73.377 102.572   
 73.968 101.663    74.548 100.743    75.114  99.809    75.666  98.861   
 76.202  97.897    76.720  96.915    77.217  95.912    77.690  94.885   
 78.137  93.832    78.552  92.747    78.931  91.626    79.267  90.462   
 79.554  89.249    79.780  87.975    79.934  86.629    79.999  85.194   
 79.954  83.650    79.803  82.188    79.568  80.819    79.264  79.525   
 78.905  78.293    78.501  77.112    78.058  75.974    77.582  74.873   
 77.079  73.804    76.553  72.762    76.005  71.743    75.440  70.746   
 74.859  69.767    74.264  68.804    73.657  67.855    73.039  66.920   
 72.411  65.996    71.775  65.083    71.130  64.180    70.478  63.285   
 69.820  62.398    69.156  61.519    68.486  60.646    67.812  59.779   
 67.133  58.918    66.449  58.062    65.762  57.211    65.072  56.365   
 64.378  55.523    63.681  54.684    62.982  53.850   ] scurve 
136.287 115.278 moveto 
[ 136.287 115.278   135.622 114.477   134.959 113.673   134.301 112.866   
133.646 112.056   132.995 111.243   132.349 110.426   131.708 109.605   
131.072 108.779   130.441 107.950   129.817 107.115   129.199 106.275   
128.589 105.428   127.986 104.576   127.391 103.716   126.806 102.848   
126.231 101.972   125.667 101.087   125.115 100.190   124.576  99.283   
124.053  98.362   123.546  97.426   123.058  96.474   122.591  95.504   
122.147  94.513   121.730  93.497   121.344  92.454   120.993  91.379   
120.682  90.268   120.419  89.113   120.210  87.908   120.067  86.642   
120.002  85.304   120.031  83.878   120.155  82.502   120.357  81.204   
120.623  79.970   120.942  78.789   121.307  77.654   121.709  76.556   
122.145  75.492   122.609  74.456   123.098  73.445   123.609  72.455   
124.138  71.485   124.685  70.532   125.246  69.593   125.821  68.668   
126.408  67.755   127.006  66.853   127.613  65.960   128.230  65.077   
128.854  64.201   129.486  63.333   130.125  62.472   130.770  61.617   
131.420  60.767   132.076  59.923   132.736  59.083   133.401  58.248   
134.070  57.417   134.743  56.590   135.420  55.767   136.100  54.947   
136.783  54.130   ] scurve 
 79.881 115.337 moveto 
[  79.881 115.337    80.270 114.228    80.651 113.112    81.023 111.987   
 81.386 110.852    81.736 109.706    82.074 108.548    82.397 107.375   
 82.703 106.186    82.989 104.978    83.252 103.748    83.487 102.490   
 83.688 101.201    83.848  99.873    83.957  98.496    84.000  97.057   
 83.955  95.537    83.816  94.100    83.590  92.749    83.281  91.478   
 82.899  90.280    82.450  89.149    81.942  88.079    81.381  87.062   
 80.773  86.093    80.124  85.167    79.439  84.280    78.722  83.425   
 77.978  82.601    77.208  81.804    76.417  81.031    75.606  80.278   
 74.778  79.545    73.936  78.829    73.079  78.128    72.211  77.441   
 71.332  76.766    70.443  76.103    69.545  75.450    68.640  74.806   
 67.727  74.171    66.809  73.543    65.884  72.923    64.954  72.309   
 64.019  71.701    63.080  71.099    62.136  70.501    61.189  69.909   
 60.239  69.321    59.285  68.737    58.329  68.157    57.369  67.580   
 56.408  67.007    55.444  66.437    54.478  65.869   ] scurve 
145.023 103.838 moveto 
[ 145.023 103.838   144.099 103.293   143.178 102.747   142.260 102.197   
141.345 101.646   140.434 101.092   139.526 100.534   138.623  99.974   
137.723  99.410   136.828  98.843   135.938  98.271   135.053  97.696   
134.173  97.115   133.300  96.530   132.433  95.940   131.573  95.343   
130.720  94.740   129.875  94.131   129.040  93.513   128.214  92.887   
127.398  92.252   126.594  91.607   125.802  90.951   125.024  90.283   
124.261  89.600   123.514  88.903   122.786  88.189   122.077  87.457   
121.391  86.704   120.729  85.928   120.095  85.126   119.491  84.295   
118.920  83.432   118.386  82.533   117.894  81.594   117.447  80.610   
117.050  79.577   116.709  78.488   116.428  77.338   116.214  76.121   
116.071  74.832   116.004  73.464   116.019  72.013   116.109  70.585   
116.256  69.219   116.448  67.900   116.676  66.619   116.934  65.369   
117.216  64.145   117.519  62.942   117.839  61.758   118.175  60.590   
118.525  59.436   118.886  58.294   119.257  57.162   119.637  56.040   
120.026  54.926   120.422  53.820   ] scurve 
120.109 115.308 moveto 
[ 120.109 115.308   119.720 114.199   119.339 113.082   118.967 111.957   
118.605 110.822   118.255 109.676   117.917 108.517   117.594 107.344   
117.289 106.154   117.003 104.946   116.741 103.715   116.507 102.457   
116.307 101.167   116.148  99.837   116.041  98.459   116.000  97.018   
116.047  95.495   116.189  94.061   116.418  92.712   116.728  91.443   
117.113  90.248   117.563  89.118   118.073  88.049   118.635  87.034   
119.245  86.066   119.894  85.142   120.581  84.255   121.298  83.402   
122.044  82.578   122.814  81.782   123.606  81.009   124.417  80.257   
125.246  79.524   126.089  78.808   126.946  78.108   127.814  77.421   
128.694  76.747   129.583  76.084   130.481  75.431   131.387  74.787   
132.299  74.152   133.218  73.525   134.143  72.905   135.074  72.291   
136.009  71.683   136.948  71.081   137.892  70.484   138.839  69.891   
139.790  69.303   140.743  68.720   141.700  68.140   142.660  67.563   
143.621  66.990   144.585  66.419   145.552  65.852   ] scurve 
 54.983 103.835 moveto 
[  54.983 103.835    55.906 103.290    56.827 102.744    57.745 102.194   
 58.660 101.643    59.571 101.089    60.478 100.531    61.382  99.971   
 62.281  99.407    63.176  98.840    64.066  98.269    64.951  97.693   
 65.831  97.113    66.704  96.528    67.571  95.937    68.431  95.341   
 69.284  94.738    70.128  94.128    70.964  93.511    71.790  92.885   
 72.605  92.250    73.410  91.605    74.201  90.948    74.979  90.280   
 75.742  89.598    76.489  88.901    77.217  88.187    77.925  87.454   
 78.612  86.701    79.273  85.925    79.907  85.123    80.511  84.292   
 81.082  83.429    81.616  82.530    82.108  81.591    82.555  80.607   
 82.951  79.573    83.292  78.484    83.572  77.334    83.787  76.117   
 83.929  74.827    83.996  73.459    83.981  72.008    83.891  70.581   
 83.744  69.214    83.552  67.895    83.323  66.614    83.066  65.365   
 82.783  64.141    82.480  62.938    82.159  61.754    81.823  60.586   
 81.474  59.432    81.113  58.290    80.742  57.158    80.361  56.036   
 79.973  54.922    79.577  53.816   ] scurve 
144.934  92.262 moveto 
[ 144.934  92.262   143.754  91.979   142.578  91.694   141.404  91.407   
140.234  91.117   139.068  90.825   137.905  90.530   136.747  90.233   
135.593  89.932   134.444  89.627   133.300  89.318   132.162  89.006   
131.031  88.688   129.906  88.365   128.789  88.036   127.679  87.701   
126.579  87.358   125.489  87.008   124.410  86.648   123.342  86.278   
122.288  85.897   121.249  85.503   120.227  85.094   119.223  84.669   
118.240  84.224   117.280  83.759   116.346  83.269   115.441  82.752   
114.569  82.203   113.732  81.619   112.937  80.995   112.185  80.326   
111.483  79.608   110.834  78.834   110.242  78.001   109.711  77.104   
109.242  76.140   108.839  75.106   108.500  74.002   108.225  72.829   
108.011  71.589   107.855  70.285   107.754  68.922   107.702  67.505   
107.695  66.039   107.727  64.556   107.790  63.108   107.877  61.685   
107.984  60.284   108.106  58.900   108.241  57.530   108.388  56.172   
108.544  54.825   108.708  53.486   ] scurve 
 55.181  92.235 moveto 
[  55.181  92.235    56.360  91.952    57.536  91.666    58.709  91.379   
 59.878  91.089    61.044  90.797    62.206  90.502    63.364  90.204   
 64.517  89.903    65.665  89.598    66.808  89.289    67.945  88.976   
 69.076  88.658    70.200  88.334    71.316  88.005    72.425  87.669   
 73.524  87.326    74.613  86.974    75.691  86.614    76.757  86.243   
 77.810  85.861    78.847  85.465    79.868  85.055    80.870  84.628   
 81.851  84.182    82.808  83.714    83.740  83.222    84.642  82.702   
 85.511  82.150    86.343  81.563    87.135  80.935    87.882  80.262   
 88.579  79.538    89.223  78.759    89.810  77.921    90.335  77.017   
 90.798  76.047    91.195  75.006    91.528  73.896    91.798  72.717   
 92.006  71.471    92.156  70.161    92.253  68.793    92.301  67.371   
 92.304  65.900    92.268  64.421    92.203  62.976    92.114  61.556   
 92.006  60.156    91.882  58.773    91.746  57.405    91.598  56.048   
 91.441  54.701    91.276  53.363   ] scurve 
 91.436 115.344 moveto 
[  91.436 115.344    91.592 114.005    91.739 112.656    91.875 111.297   
 91.999 109.926    92.107 108.541    92.197 107.139    92.263 105.716   
 92.302 104.267    92.303 102.785    92.258 101.300    92.159  99.864   
 91.999  98.482    91.773  97.160    91.475  95.904    91.099  94.718   
 90.642  93.604    90.105  92.565    89.488  91.600    88.794  90.707   
 88.029  89.882    87.199  89.119    86.311  88.413    85.373  87.758   
 84.390  87.148    83.370  86.578    82.318  86.042    81.238  85.536   
 80.135  85.056    79.012  84.599    77.871  84.162    76.717  83.743   
 75.550  83.338    74.372  82.947    73.185  82.567    71.990  82.198   
 70.788  81.838    69.580  81.486    68.367  81.142    67.149  80.804   
 65.928  80.473    64.702  80.146    63.474  79.825    62.243  79.507   
 61.009  79.194    59.773  78.885    58.535  78.578    57.296  78.275   
 56.054  77.975    54.812  77.677   ] scurve 
108.558 115.294 moveto 
[ 108.558 115.294   108.402 113.953   108.255 112.604   108.120 111.245   
107.997 109.874   107.889 108.488   107.800 107.085   107.735 105.662   
107.698 104.211   107.698 102.728   107.745 101.244   107.846  99.810   
108.008  98.431   108.237  97.111   108.538  95.857   108.917  94.674   
109.377  93.563   109.917  92.527   110.537  91.565   111.234  90.675   
112.002  89.852   112.834  89.091   113.724  88.387   114.664  87.734   
115.648  87.126   116.670  86.557   117.723  86.022   118.804  85.517   
119.908  85.038   121.032  84.582   122.173  84.146   123.329  83.727   
124.496  83.322   125.675  82.932   126.862  82.552   128.057  82.184   
129.260  81.824   130.468  81.473   131.682  81.129   132.900  80.791   
134.122  80.459   135.347  80.133   136.576  79.812   137.808  79.495   
139.042  79.181   140.278  78.872   141.516  78.566   142.756  78.263   
143.997  77.962   145.240  77.665   ] scurve 
grestore}
\LabelTeX   148.000  81.200 $x$\ELTX
\LabelTeX    96.500 117.500 $y$\ELTX
\EndFig
\vskip-45mm
}

\subsection 3.3. Courbe de poursuite du chien|

Nous pr\'esentons ici la c\'el\`ebre \og$\,$courbe du chien$\,$\fg\ comme
exemple de courbe de poursuite. Voici le probl\`eme~: un chien et son
ma\^{\i}tre se d\'eplacent l'un et l'autre \`a des vitesses scalaires
constantes $V$ (pour le chien) et $v$ (pour le ma\^{\i}tre), avec $V>v$. 
On suppose que le ma\^{\i}tre se d\'eplace en ligne droite, disons sur
l'axe $Ox$, dans la direction positive, suivant la loi $x=vt$. A
l'instant $t=0$, le chien se trouve au point $x=0$ $y=r_0$, \`a distance
$r_0$ du ma\^{\i}tre. Le chien cherche \`a rejoindre son ma\^{\i}tre en
pointant son vecteur vitesse $\overrightarrow V$ en direction du
ma\^{\i}tre. Le probl\`eme est de d\'eterminer la loi du mouvement
$C(t)$ du chien.

\InsertFig 18.000 65.000 {
1 mm unit
gsave 
  0.000   0.000 moveto 
100.000   0.000 vector 
 60.000  90.000 vector
  0.000   0.000 moveto
  0.700  disk
 45.000   0.000 moveto
  0.700  disk
  0.000  50.000 moveto
  0.700  disk
 20.000  16.900 moveto
  0.700  disk
  0.3 setlinewidth
 80.000   0.000 moveto
  1.000   0.000 cross
 80.000 50.000  moveto
 80.000 50.000  180.000 270.000 ellipsearc stroke
  0.5 setlinewidth
 45.000   0.000 moveto
 10.000   0.000 vector
  0.000  50.000 moveto
 17.000 -90.000 vector
 20.000  16.900 moveto
 17.000  -34.200 vector
  0.113 setlinewidth
 [ 0.75 0.5 ] 0 setdash
  20.000  16.900 moveto
  45.000   0.000 lineto stroke
 [ ] 0 setdash
 45.000 0.000 moveto
  5.000 5.000  145.800 180.000 ellipsearc stroke  
 45.000 0.000 moveto
  5.700 5.700  145.800 180.000 ellipsearc stroke  
grestore}
\LabelTeX   -5.800 32.500 $\overrightarrow{V}$\ELTX
\LabelTeX   27.500  5.000 $\overrightarrow{V}$\ELTX
\LabelTeX   -6.800 -4.500 $M_0=O$\ELTX
\LabelTeX   41.000 -6.500 $\displaystyle M(t)\pmatrix{vt\cr 0\cr}$\ELTX
\LabelTeX   98.000 -3.500 $x$\ELTX
\LabelTeX   -4.000 58.500 $y$\ELTX
\LabelTeX   36.500  1.000 $\alpha$\ELTX
\LabelTeX    1.800 49.500 $\displaystyle C_0\pmatrix{0\cr r_0\cr}$\ELTX
\LabelTeX   20.000 19.000 $\displaystyle C(t)\pmatrix{x(t)\cr y(t)\cr}$\ELTX
\EndFig
\vskip20pt

Notons $\alpha$ l'angle (non orient\'e) $\alpha=(Ox,\overrightarrow{CM})$ et
$r=\Vert\overrightarrow{CM}\Vert\;$; on a bien entendu 
$\alpha=\alpha(t)$ et $r=r(t)$. Comme d'habitude en Physique, on d\'esignera 
par des points surlignants les d\'eriv\'ees  temporelles $\dot r(t)=dr/dt$, 
$\dot\alpha(t)=d\alpha/dt,\,\ldots\;.$ A l'instant $t$, la position et 
la vitesse du chien sont donn\'ees~par
$$
\bigg\{\cmalign{
x(t)&{}=vt-r\cos\alpha,\qquad \dot x(t)&{}=v-\dot r\cos\alpha+r\dot\alpha
\sin\alpha=V\cos\alpha,\cr
\noalign{\vskip3pt}
y(t)&{}=r\sin\alpha,\qquad\hfill
\dot y(t)&{}=\dot r\sin\alpha+r\dot\alpha\cos\alpha=-V\sin\alpha,\cr
}
$$
Ces \'equations fournissent ais\'ement l'expression de
$\dot r$ et $r\dot\alpha$~:
$$
\bigg\{\eqalign{
\dot r&= v\cos\alpha-V,\cr
r\dot \alpha&= -v\sin\alpha.\cr}
$$
En prenant le quotient on \'elimine $dt$ et on trouve donc
$${dr\over r\,d\alpha}=-{\rm cotan}\,\alpha+{V\over v}{1\over \sin\alpha}.$$
Notons $\lambda=V/v>1$ le rapport des vitesses respectives du chien et du 
ma\^{\i}tre. Apr\`es int\'egration, et compte tenu de ce que
$r=r_0$ et $\alpha=\pi/2$ quand $t=0$, il vient
$$
\ln r =  -\ln\sin\alpha+\lambda\ln\tan(\alpha/2)+\hbox{Cte}\quad
\Longrightarrow\quad
r=r_0{\tan(\alpha/2)^{\lambda}\over \sin\alpha}
$$
Nous en d\'eduisons
$$
{d\alpha\over dt}=\dot\alpha=-{v\sin\alpha\over r}=
-{v\over r_0}(\sin\alpha)^2\tan(\alpha/2)^{-\lambda}.
$$
En posant $\theta=\tan(\alpha/2)$
et $\displaystyle \sin\alpha=2\sin{\alpha\over 2}\cos{\alpha\over 2}=
{2\theta\over 1+\theta^2}$, on trouve
$$
\eqalign{
dt&=-{r_0\over v}\;{\tan(\alpha/2)^{\lambda}\over(\sin\alpha)^2}\,d\alpha
=-{r_0\over 2v}(1+\theta^2)\theta^{\lambda-2}d\theta,\cr
t&={r_0\over 2v}\Big({1-\theta^{\lambda-1}\over \lambda-1}+
{1-\theta^{\lambda+1}\over \lambda+1}\Big)\cr}
$$
compte tenu du fait que $\theta=1$ en $t=0$. Par substitution dans
les expressions de $x$ et $y$, et d'apr\`es l'\'egalit\'e
$\tan\alpha=2\theta/(1-\theta^2)$, on obtient les \'equations
param\'etriques de la \og$\,$courbe du chien$\,$\fg, \`a savoir
$$
\cases{\displaystyle
x={r_0\over 2}\Big({1-\theta^{\lambda-1}\over \lambda-1}+
{1-\theta^{\lambda+1}\over \lambda+1}\Big)-{r_0\over 2}(1-\theta^2)\theta^{\lambda-1}&\cr
\noalign{\vskip4pt}
y=r_0\theta^{\lambda}&\cr
\noalign{\vskip4pt}
\displaystyle
t={r_0\over 2v}\Big({1-\theta^{\lambda-1}\over \lambda-1}+
{1-\theta^{\lambda+1}\over \lambda+1}\Big),\qquad \theta\in[0,1].&\cr}
$$
Au terme de la poursuite ($y=\theta=0$), le ma\^{\i}tre a parcouru la
distance
$$
x={\lambda\over \lambda^2-1}\,r_0\qquad
\hbox{pendant le temps}\quad
t={\lambda\over \lambda^2-1}\,{r_0\over v}.\qquad
$$

\InsertFig 28.000 75.000 {
1 mm unit
gsave 
-10.000   0.000 moveto 
100.000   0.000 vector 
  0.000   0.000 moveto 
 73.000  90.000 vector
  0.000  35.000 moveto
  0.700 disk
 0.226 setlinewidth
 11.121  70.000  moveto
[ 11.121  70.000    11.020  69.825    10.918  69.650    10.817  69.475  
  10.716  69.300    10.616  69.125    10.516  68.950    10.417  68.775  
  10.318  68.600    10.219  68.425    10.121  68.250    10.023  68.075  
   9.926  67.900     9.829  67.725     9.732  67.550     9.636  67.375  
   9.540  67.200     9.445  67.025     9.350  66.850     9.256  66.675  
   9.161  66.500     9.068  66.325     8.975  66.150     8.882  65.975  
   8.789  65.800     8.697  65.625     8.606  65.450     8.515  65.275  
   8.424  65.100     8.334  64.925     8.244  64.750     8.154  64.575  
   8.065  64.400     7.977  64.225     7.889  64.050     7.801  63.875  
   7.714  63.700     7.627  63.525     7.540  63.350     7.454  63.175  
   7.369  63.000     7.284  62.825     7.199  62.650     7.115  62.475  
   7.031  62.300     6.947  62.125     6.864  61.950     6.782  61.775  
   6.700  61.600     6.618  61.425     6.537  61.250     6.456  61.075  
   6.376  60.900     6.296  60.725     6.217  60.550     6.138  60.375  
   6.059  60.200     5.981  60.025     5.903  59.850     5.826  59.675  
   5.749  59.500     5.673  59.325     5.597  59.150     5.522  58.975  
   5.447  58.800     5.372  58.625     5.298  58.450     5.225  58.275  
   5.151  58.100     5.079  57.925     5.007  57.750     4.935  57.575  
   4.863  57.400     4.793  57.225     4.722  57.050     4.652  56.875  
   4.583  56.700     4.514  56.525     4.445  56.350     4.377  56.175  
   4.310  56.000     4.243  55.825     4.176  55.650     4.110  55.475  
   4.044  55.300     3.979  55.125     3.914  54.950     3.850  54.775  
   3.786  54.600     3.722  54.425     3.660  54.250     3.597  54.075  
   3.535  53.900     3.474  53.725     3.413  53.550     3.353  53.375  
   3.293  53.200     3.233  53.025     3.174  52.850     3.116  52.675  
   3.058  52.500     3.000  52.325     2.943  52.150     2.887  51.975  
   2.830  51.800     2.775  51.625     2.720  51.450     2.665  51.275  
   2.611  51.100     2.558  50.925     2.505  50.750     2.452  50.575  
   2.400  50.400     2.349  50.225     2.298  50.050     2.247  49.875  
   2.197  49.700     2.148  49.525     2.099  49.350     2.050  49.175  
   2.002  49.000     1.955  48.825     1.908  48.650     1.862  48.475  
   1.816  48.300     1.770  48.125     1.726  47.950     1.681  47.775  
   1.638  47.600     1.594  47.425     1.552  47.250     1.510  47.075  
   1.468  46.900     1.427  46.725     1.386  46.550     1.346  46.375  
   1.307  46.200     1.268  46.025     1.230  45.850     1.192  45.675  
   1.154  45.500     1.118  45.325     1.082  45.150     1.046  44.975  
   1.011  44.800     0.976  44.625     0.942  44.450     0.909  44.275  
   0.876  44.100     0.844  43.925     0.812  43.750     0.781  43.575  
   0.751  43.400     0.721  43.225     0.691  43.050     0.662  42.875  
   0.634  42.700     0.606  42.525     0.579  42.350     0.553  42.175  
   0.527  42.000     0.502  41.825     0.477  41.650     0.453  41.475  
   0.429  41.300     0.406  41.125     0.384  40.950     0.362  40.775  
   0.341  40.600     0.321  40.425     0.301  40.250     0.281  40.075  
   0.263  39.900     0.245  39.725     0.227  39.550     0.210  39.375  
   0.194  39.200     0.179  39.025     0.164  38.850     0.149  38.675  
   0.136  38.500     0.123  38.325     0.110  38.150     0.098  37.975  
   0.087  37.800     0.077  37.625     0.067  37.450     0.058  37.275  
   0.049  37.100     0.042  36.925     0.034  36.750     0.028  36.575  
   0.022  36.400     0.017  36.225     0.012  36.050     0.009  35.875  
   0.006  35.700     0.003  35.525     0.001  35.350     0.000  35.175  
   0.000  35.000     0.000  34.825     0.001  34.650     0.003  34.475  
   0.006  34.300     0.009  34.125     0.013  33.950     0.017  33.775  
   0.023  33.600     0.029  33.425     0.036  33.250     0.043  33.075  
   0.051  32.900     0.060  32.725     0.070  32.550     0.081  32.375  
   0.092  32.200     0.104  32.025     0.117  31.850     0.131  31.675  
   0.145  31.500     0.160  31.325     0.176  31.150     0.193  30.975  
   0.210  30.800     0.229  30.625     0.248  30.450     0.268  30.275  
   0.289  30.100     0.310  29.925     0.333  29.750     0.356  29.575  
   0.380  29.400     0.405  29.225     0.430  29.050     0.457  28.875  
   0.485  28.700     0.513  28.525     0.542  28.350     0.572  28.175  
   0.603  28.000     0.635  27.825     0.668  27.650     0.701  27.475  
   0.736  27.300     0.771  27.125     0.808  26.950     0.845  26.775  
   0.883  26.600     0.923  26.425     0.963  26.250     1.004  26.075  
   1.046  25.900     1.089  25.725     1.133  25.550     1.178  25.375  
   1.224  25.200     1.271  25.025     1.319  24.850     1.368  24.675  
   1.418  24.500     1.469  24.325     1.522  24.150     1.575  23.975  
   1.629  23.800     1.685  23.625     1.741  23.450     1.799  23.275  
   1.857  23.100     1.917  22.925     1.978  22.750     2.040  22.575  
   2.103  22.400     2.168  22.225     2.233  22.050     2.300  21.875  
   2.368  21.700     2.437  21.525     2.508  21.350     2.579  21.175  
   2.652  21.000     2.726  20.825     2.802  20.650     2.879  20.475  
   2.957  20.300     3.036  20.125     3.117  19.950     3.199  19.775  
   3.282  19.600     3.367  19.425     3.453  19.250     3.541  19.075  
   3.630  18.900     3.720  18.725     3.812  18.550     3.906  18.375  
   4.001  18.200     4.097  18.025     4.196  17.850     4.295  17.675  
   4.397  17.500     4.499  17.325     4.604  17.150     4.710  16.975  
   4.818  16.800     4.928  16.625     5.039  16.450     5.152  16.275  
   5.267  16.100     5.384  15.925     5.503  15.750     5.623  15.575  
   5.746  15.400     5.870  15.225     5.996  15.050     6.125  14.875  
   6.255  14.700     6.387  14.525     6.522  14.350     6.659  14.175  
   6.798  14.000     6.939  13.825     7.082  13.650     7.228  13.475  
   7.377  13.300     7.527  13.125     7.680  12.950     7.836  12.775  
   7.994  12.600     8.155  12.425     8.318  12.250     8.485  12.075  
   8.654  11.900     8.826  11.725     9.001  11.550     9.178  11.375  
   9.359  11.200     9.544  11.025     9.731  10.850     9.922  10.675  
  10.116  10.500    10.313  10.325    10.515  10.150    10.719   9.975  
  10.928   9.800    11.141   9.625    11.358   9.450    11.578   9.275  
  11.803   9.100    12.033   8.925    12.267   8.750    12.506   8.575  
  12.749   8.400    12.998   8.225    13.252   8.050    13.511   7.875  
  13.776   7.700    14.047   7.525    14.324   7.350    14.607   7.175  
  14.896   7.000    15.193   6.825    15.496   6.650    15.807   6.475  
  16.126   6.300    16.453   6.125    16.788   5.950    17.133   5.775  
  17.487   5.600    17.851   5.425    18.225   5.250    18.611   5.075  
  19.008   4.900    19.418   4.725    19.842   4.550    20.280   4.375  
  20.733   4.200    21.202   4.025    21.689   3.850    22.196   3.675  
  22.723   3.500    23.273   3.325    23.848   3.150    24.450   2.975  
  25.082   2.800    25.748   2.625    26.451   2.450    27.197   2.275  
  27.992   2.100    28.843   1.925    29.760   1.750    30.755   1.575  
  31.843   1.400    33.048   1.225    34.401   1.050    35.950   0.875  
  37.772   0.700    40.006   0.525    42.946   0.350    47.453   0.175  
  77.778   0.000    ] scurve
  77.778   0.000  moveto
 0.700 disk
  9.190  70.000  moveto
[  9.190  70.000     9.106  69.825     9.023  69.650     8.940  69.475  
   8.857  69.300     8.775  69.125     8.693  68.950     8.611  68.775  
   8.530  68.600     8.449  68.425     8.368  68.250     8.288  68.075  
   8.208  67.900     8.128  67.725     8.049  67.550     7.970  67.375  
   7.891  67.200     7.813  67.025     7.735  66.850     7.657  66.675  
   7.580  66.500     7.502  66.325     7.426  66.150     7.349  65.975  
   7.273  65.800     7.198  65.625     7.122  65.450     7.047  65.275  
   6.973  65.100     6.898  64.925     6.825  64.750     6.751  64.575  
   6.678  64.400     6.605  64.225     6.532  64.050     6.460  63.875  
   6.388  63.700     6.316  63.525     6.245  63.350     6.174  63.175  
   6.104  63.000     6.034  62.825     5.964  62.650     5.894  62.475  
   5.825  62.300     5.756  62.125     5.688  61.950     5.620  61.775  
   5.552  61.600     5.485  61.425     5.418  61.250     5.351  61.075  
   5.285  60.900     5.219  60.725     5.153  60.550     5.088  60.375  
   5.023  60.200     4.959  60.025     4.895  59.850     4.831  59.675  
   4.768  59.500     4.705  59.325     4.642  59.150     4.580  58.975  
   4.518  58.800     4.456  58.625     4.395  58.450     4.334  58.275  
   4.274  58.100     4.214  57.925     4.154  57.750     4.095  57.575  
   4.036  57.400     3.977  57.225     3.919  57.050     3.861  56.875  
   3.804  56.700     3.747  56.525     3.690  56.350     3.634  56.175  
   3.578  56.000     3.522  55.825     3.467  55.650     3.412  55.475  
   3.358  55.300     3.304  55.125     3.250  54.950     3.197  54.775  
   3.144  54.600     3.092  54.425     3.040  54.250     2.988  54.075  
   2.937  53.900     2.886  53.725     2.835  53.550     2.785  53.375  
   2.736  53.200     2.686  53.025     2.637  52.850     2.589  52.675  
   2.541  52.500     2.493  52.325     2.446  52.150     2.399  51.975  
   2.353  51.800     2.306  51.625     2.261  51.450     2.216  51.275  
   2.171  51.100     2.126  50.925     2.082  50.750     2.039  50.575  
   1.996  50.400     1.953  50.225     1.911  50.050     1.869  49.875  
   1.827  49.700     1.786  49.525     1.745  49.350     1.705  49.175  
   1.665  49.000     1.626  48.825     1.587  48.650     1.549  48.475  
   1.510  48.300     1.473  48.125     1.436  47.950     1.399  47.775  
   1.362  47.600     1.327  47.425     1.291  47.250     1.256  47.075  
   1.221  46.900     1.187  46.725     1.154  46.550     1.120  46.375  
   1.088  46.200     1.055  46.025     1.023  45.850     0.992  45.675  
   0.961  45.500     0.930  45.325     0.900  45.150     0.871  44.975  
   0.842  44.800     0.813  44.625     0.785  44.450     0.757  44.275  
   0.730  44.100     0.703  43.925     0.676  43.750     0.650  43.575  
   0.625  43.400     0.600  43.225     0.576  43.050     0.552  42.875  
   0.528  42.700     0.505  42.525     0.483  42.350     0.460  42.175  
   0.439  42.000     0.418  41.825     0.397  41.650     0.377  41.475  
   0.358  41.300     0.338  41.125     0.320  40.950     0.302  40.775  
   0.284  40.600     0.267  40.425     0.250  40.250     0.234  40.075  
   0.219  39.900     0.204  39.725     0.189  39.550     0.175  39.375  
   0.162  39.200     0.149  39.025     0.136  38.850     0.124  38.675  
   0.113  38.500     0.102  38.325     0.092  38.150     0.082  37.975  
   0.073  37.800     0.064  37.625     0.056  37.450     0.048  37.275  
   0.041  37.100     0.035  36.925     0.029  36.750     0.023  36.575  
   0.018  36.400     0.014  36.225     0.010  36.050     0.007  35.875  
   0.005  35.700     0.003  35.525     0.001  35.350     0.000  35.175  
   0.000  35.000     0.000  34.825     0.001  34.650     0.003  34.475  
   0.005  34.300     0.007  34.125     0.011  33.950     0.014  33.775  
   0.019  33.600     0.024  33.425     0.030  33.250     0.036  33.075  
   0.043  32.900     0.050  32.725     0.059  32.550     0.067  32.375  
   0.077  32.200     0.087  32.025     0.098  31.850     0.109  31.675  
   0.121  31.500     0.133  31.325     0.147  31.150     0.161  30.975  
   0.175  30.800     0.191  30.625     0.206  30.450     0.223  30.275  
   0.240  30.100     0.258  29.925     0.277  29.750     0.296  29.575  
   0.316  29.400     0.337  29.225     0.359  29.050     0.381  28.875  
   0.404  28.700     0.427  28.525     0.451  28.350     0.476  28.175  
   0.502  28.000     0.529  27.825     0.556  27.650     0.584  27.475  
   0.613  27.300     0.642  27.125     0.672  26.950     0.703  26.775  
   0.735  26.600     0.768  26.425     0.801  26.250     0.835  26.075  
   0.870  25.900     0.906  25.725     0.943  25.550     0.980  25.375  
   1.018  25.200     1.057  25.025     1.097  24.850     1.138  24.675  
   1.180  24.500     1.222  24.325     1.265  24.150     1.309  23.975  
   1.355  23.800     1.400  23.625     1.447  23.450     1.495  23.275  
   1.544  23.100     1.593  22.925     1.644  22.750     1.695  22.575  
   1.748  22.400     1.801  22.225     1.855  22.050     1.910  21.875  
   1.967  21.700     2.024  21.525     2.082  21.350     2.141  21.175  
   2.201  21.000     2.263  20.825     2.325  20.650     2.388  20.475  
   2.453  20.300     2.518  20.125     2.585  19.950     2.653  19.775  
   2.721  19.600     2.791  19.425     2.862  19.250     2.935  19.075  
   3.008  18.900     3.082  18.725     3.158  18.550     3.235  18.375  
   3.313  18.200     3.392  18.025     3.473  17.850     3.555  17.675  
   3.638  17.500     3.722  17.325     3.808  17.150     3.895  16.975  
   3.984  16.800     4.073  16.625     4.165  16.450     4.257  16.275  
   4.351  16.100     4.447  15.925     4.543  15.750     4.642  15.575  
   4.742  15.400     4.843  15.225     4.946  15.050     5.051  14.875  
   5.157  14.700     5.264  14.525     5.374  14.350     5.485  14.175  
   5.598  14.000     5.712  13.825     5.829  13.650     5.947  13.475  
   6.067  13.300     6.189  13.125     6.312  12.950     6.438  12.775  
   6.566  12.600     6.695  12.425     6.827  12.250     6.961  12.075  
   7.097  11.900     7.235  11.725     7.375  11.550     7.518  11.375  
   7.662  11.200     7.810  11.025     7.959  10.850     8.112  10.675  
   8.266  10.500     8.424  10.325     8.584  10.150     8.747   9.975  
   8.912   9.800     9.081   9.625     9.252   9.450     9.426   9.275  
   9.604   9.100     9.785   8.925     9.969   8.750    10.156   8.575  
  10.347   8.400    10.542   8.225    10.740   8.050    10.942   7.875  
  11.149   7.700    11.359   7.525    11.573   7.350    11.792   7.175  
  12.016   7.000    12.244   6.825    12.478   6.650    12.716   6.475  
  12.960   6.300    13.209   6.125    13.465   5.950    13.726   5.775  
  13.994   5.600    14.268   5.425    14.550   5.250    14.839   5.075  
  15.136   4.900    15.441   4.725    15.755   4.550    16.078   4.375  
  16.411   4.200    16.755   4.025    17.110   3.850    17.477   3.675  
  17.858   3.500    18.252   3.325    18.662   3.150    19.089   2.975  
  19.534   2.800    20.000   2.625    20.488   2.450    21.002   2.275  
  21.543   2.100    22.118   1.925    22.730   1.750    23.386   1.575  
  24.094   1.400    24.866   1.225    25.717   1.050    26.671   0.875  
  27.765   0.700    29.062   0.525    30.694   0.350    33.024   0.175  
  42.000   0.000    ] scurve
  42.000   0.000  moveto
 0.700 disk
  6.834  70.000  moveto
[  6.834  70.000     6.772  69.825     6.711  69.650     6.650  69.475  
   6.589  69.300     6.528  69.125     6.467  68.950     6.407  68.775  
   6.347  68.600     6.287  68.425     6.227  68.250     6.168  68.075  
   6.109  67.900     6.050  67.725     5.991  67.550     5.932  67.375  
   5.874  67.200     5.816  67.025     5.759  66.850     5.701  66.675  
   5.644  66.500     5.587  66.325     5.530  66.150     5.474  65.975  
   5.417  65.800     5.361  65.625     5.305  65.450     5.250  65.275  
   5.195  65.100     5.139  64.925     5.085  64.750     5.030  64.575  
   4.976  64.400     4.922  64.225     4.868  64.050     4.814  63.875  
   4.761  63.700     4.708  63.525     4.655  63.350     4.603  63.175  
   4.550  63.000     4.498  62.825     4.447  62.650     4.395  62.475  
   4.344  62.300     4.293  62.125     4.242  61.950     4.191  61.775  
   4.141  61.600     4.091  61.425     4.041  61.250     3.992  61.075  
   3.943  60.900     3.894  60.725     3.845  60.550     3.797  60.375  
   3.748  60.200     3.701  60.025     3.653  59.850     3.606  59.675  
   3.558  59.500     3.512  59.325     3.465  59.150     3.419  58.975  
   3.373  58.800     3.327  58.625     3.281  58.450     3.236  58.275  
   3.191  58.100     3.147  57.925     3.102  57.750     3.058  57.575  
   3.014  57.400     2.970  57.225     2.927  57.050     2.884  56.875  
   2.841  56.700     2.799  56.525     2.757  56.350     2.715  56.175  
   2.673  56.000     2.632  55.825     2.591  55.650     2.550  55.475  
   2.509  55.300     2.469  55.125     2.429  54.950     2.390  54.775  
   2.350  54.600     2.311  54.425     2.272  54.250     2.234  54.075  
   2.196  53.900     2.158  53.725     2.120  53.550     2.083  53.375  
   2.046  53.200     2.009  53.025     1.972  52.850     1.936  52.675  
   1.900  52.500     1.865  52.325     1.829  52.150     1.794  51.975  
   1.760  51.800     1.725  51.625     1.691  51.450     1.658  51.275  
   1.624  51.100     1.591  50.925     1.558  50.750     1.526  50.575  
   1.493  50.400     1.461  50.225     1.430  50.050     1.398  49.875  
   1.367  49.700     1.337  49.525     1.306  49.350     1.276  49.175  
   1.247  49.000     1.217  48.825     1.188  48.650     1.159  48.475  
   1.131  48.300     1.103  48.125     1.075  47.950     1.047  47.775  
   1.020  47.600     0.993  47.425     0.967  47.250     0.941  47.075  
   0.915  46.900     0.889  46.725     0.864  46.550     0.839  46.375  
   0.815  46.200     0.790  46.025     0.767  45.850     0.743  45.675  
   0.720  45.500     0.697  45.325     0.674  45.150     0.652  44.975  
   0.630  44.800     0.609  44.625     0.588  44.450     0.567  44.275  
   0.547  44.100     0.527  43.925     0.507  43.750     0.487  43.575  
   0.468  43.400     0.450  43.225     0.431  43.050     0.413  42.875  
   0.396  42.700     0.379  42.525     0.362  42.350     0.345  42.175  
   0.329  42.000     0.313  41.825     0.298  41.650     0.283  41.475  
   0.268  41.300     0.254  41.125     0.240  40.950     0.226  40.775  
   0.213  40.600     0.200  40.425     0.188  40.250     0.176  40.075  
   0.164  39.900     0.153  39.725     0.142  39.550     0.131  39.375  
   0.121  39.200     0.112  39.025     0.102  38.850     0.093  38.675  
   0.085  38.500     0.077  38.325     0.069  38.150     0.062  37.975  
   0.055  37.800     0.048  37.625     0.042  37.450     0.036  37.275  
   0.031  37.100     0.026  36.925     0.022  36.750     0.017  36.575  
   0.014  36.400     0.011  36.225     0.008  36.050     0.005  35.875  
   0.003  35.700     0.002  35.525     0.001  35.350     0.000  35.175  
   0.000  35.000     0.000  34.825     0.001  34.650     0.002  34.475  
   0.004  34.300     0.006  34.125     0.008  33.950     0.011  33.775  
   0.014  33.600     0.018  33.425     0.022  33.250     0.027  33.075  
   0.032  32.900     0.038  32.725     0.044  32.550     0.051  32.375  
   0.058  32.200     0.065  32.025     0.073  31.850     0.082  31.675  
   0.091  31.500     0.100  31.325     0.110  31.150     0.120  30.975  
   0.131  30.800     0.143  30.625     0.155  30.450     0.167  30.275  
   0.180  30.100     0.194  29.925     0.208  29.750     0.222  29.575  
   0.237  29.400     0.253  29.225     0.269  29.050     0.285  28.875  
   0.302  28.700     0.320  28.525     0.338  28.350     0.357  28.175  
   0.376  28.000     0.396  27.825     0.417  27.650     0.438  27.475  
   0.459  27.300     0.481  27.125     0.504  26.950     0.527  26.775  
   0.551  26.600     0.575  26.425     0.600  26.250     0.626  26.075  
   0.652  25.900     0.679  25.725     0.706  25.550     0.734  25.375  
   0.762  25.200     0.792  25.025     0.821  24.850     0.852  24.675  
   0.883  24.500     0.915  24.325     0.947  24.150     0.980  23.975  
   1.014  23.800     1.048  23.625     1.083  23.450     1.118  23.275  
   1.155  23.100     1.192  22.925     1.229  22.750     1.268  22.575  
   1.307  22.400     1.346  22.225     1.387  22.050     1.428  21.875  
   1.470  21.700     1.512  21.525     1.556  21.350     1.600  21.175  
   1.645  21.000     1.690  20.825     1.737  20.650     1.784  20.475  
   1.831  20.300     1.880  20.125     1.930  19.950     1.980  19.775  
   2.031  19.600     2.083  19.425     2.135  19.250     2.189  19.075  
   2.243  18.900     2.298  18.725     2.354  18.550     2.411  18.375  
   2.469  18.200     2.528  18.025     2.587  17.850     2.648  17.675  
   2.709  17.500     2.772  17.325     2.835  17.150     2.899  16.975  
   2.964  16.800     3.031  16.625     3.098  16.450     3.166  16.275  
   3.235  16.100     3.305  15.925     3.376  15.750     3.449  15.575  
   3.522  15.400     3.596  15.225     3.672  15.050     3.749  14.875  
   3.826  14.700     3.905  14.525     3.985  14.350     4.066  14.175  
   4.149  14.000     4.232  13.825     4.317  13.650     4.403  13.475  
   4.491  13.300     4.579  13.125     4.669  12.950     4.761  12.775  
   4.853  12.600     4.947  12.425     5.043  12.250     5.140  12.075  
   5.238  11.900     5.338  11.725     5.439  11.550     5.542  11.375  
   5.646  11.200     5.752  11.025     5.860  10.850     5.969  10.675  
   6.080  10.500     6.193  10.325     6.307  10.150     6.424   9.975  
   6.542   9.800     6.662   9.625     6.784   9.450     6.908   9.275  
   7.033   9.100     7.162   8.925     7.292   8.750     7.424   8.575  
   7.559   8.400     7.696   8.225     7.835   8.050     7.977   7.875  
   8.121   7.700     8.268   7.525     8.417   7.350     8.569   7.175  
   8.724   7.000     8.882   6.825     9.043   6.650     9.208   6.475  
   9.375   6.300     9.546   6.125     9.720   5.950     9.898   5.775  
  10.080   5.600    10.266   5.425    10.456   5.250    10.650   5.075  
  10.849   4.900    11.052   4.725    11.261   4.550    11.475   4.375  
  11.694   4.200    11.919   4.025    12.151   3.850    12.389   3.675  
  12.634   3.500    12.887   3.325    13.148   3.150    13.418   2.975  
  13.698   2.800    13.988   2.625    14.289   2.450    14.603   2.275  
  14.932   2.100    15.276   1.925    15.638   1.750    16.020   1.575  
  16.427   1.400    16.862   1.225    17.332   1.050    17.845   0.875  
  18.417   0.700    19.068   0.525    19.845   0.350    20.863   0.175  
  23.333   0.000    ] scurve
  23.333   0.000  moveto
 0.700 disk
  4.529  70.000  moveto
[  4.529  70.000     4.488  69.825     4.447  69.650     4.407  69.475  
   4.367  69.300     4.327  69.125     4.287  68.950     4.247  68.775  
   4.207  68.600     4.168  68.425     4.128  68.250     4.089  68.075  
   4.050  67.900     4.011  67.725     3.972  67.550     3.934  67.375  
   3.895  67.200     3.857  67.025     3.819  66.850     3.781  66.675  
   3.743  66.500     3.706  66.325     3.668  66.150     3.631  65.975  
   3.594  65.800     3.557  65.625     3.520  65.450     3.483  65.275  
   3.446  65.100     3.410  64.925     3.374  64.750     3.338  64.575  
   3.302  64.400     3.266  64.225     3.231  64.050     3.195  63.875  
   3.160  63.700     3.125  63.525     3.090  63.350     3.055  63.175  
   3.021  63.000     2.986  62.825     2.952  62.650     2.918  62.475  
   2.884  62.300     2.850  62.125     2.817  61.950     2.783  61.775  
   2.750  61.600     2.717  61.425     2.684  61.250     2.651  61.075  
   2.618  60.900     2.586  60.725     2.554  60.550     2.522  60.375  
   2.490  60.200     2.458  60.025     2.427  59.850     2.395  59.675  
   2.364  59.500     2.333  59.325     2.302  59.150     2.271  58.975  
   2.241  58.800     2.211  58.625     2.181  58.450     2.151  58.275  
   2.121  58.100     2.091  57.925     2.062  57.750     2.032  57.575  
   2.003  57.400     1.974  57.225     1.946  57.050     1.917  56.875  
   1.889  56.700     1.861  56.525     1.833  56.350     1.805  56.175  
   1.777  56.000     1.750  55.825     1.723  55.650     1.695  55.475  
   1.669  55.300     1.642  55.125     1.615  54.950     1.589  54.775  
   1.563  54.600     1.537  54.425     1.511  54.250     1.486  54.075  
   1.460  53.900     1.435  53.725     1.410  53.550     1.385  53.375  
   1.361  53.200     1.336  53.025     1.312  52.850     1.288  52.675  
   1.264  52.500     1.241  52.325     1.217  52.150     1.194  51.975  
   1.171  51.800     1.148  51.625     1.126  51.450     1.103  51.275  
   1.081  51.100     1.059  50.925     1.037  50.750     1.015  50.575  
   0.994  50.400     0.973  50.225     0.952  50.050     0.931  49.875  
   0.910  49.700     0.890  49.525     0.870  49.350     0.850  49.175  
   0.830  49.000     0.810  48.825     0.791  48.650     0.772  48.475  
   0.753  48.300     0.734  48.125     0.716  47.950     0.697  47.775  
   0.679  47.600     0.662  47.425     0.644  47.250     0.626  47.075  
   0.609  46.900     0.592  46.725     0.575  46.550     0.559  46.375  
   0.543  46.200     0.526  46.025     0.511  45.850     0.495  45.675  
   0.480  45.500     0.464  45.325     0.449  45.150     0.435  44.975  
   0.420  44.800     0.406  44.625     0.392  44.450     0.378  44.275  
   0.364  44.100     0.351  43.925     0.338  43.750     0.325  43.575  
   0.312  43.400     0.300  43.225     0.287  43.050     0.275  42.875  
   0.264  42.700     0.252  42.525     0.241  42.350     0.230  42.175  
   0.219  42.000     0.209  41.825     0.198  41.650     0.188  41.475  
   0.179  41.300     0.169  41.125     0.160  40.950     0.151  40.775  
   0.142  40.600     0.133  40.425     0.125  40.250     0.117  40.075  
   0.109  39.900     0.102  39.725     0.095  39.550     0.088  39.375  
   0.081  39.200     0.074  39.025     0.068  38.850     0.062  38.675  
   0.056  38.500     0.051  38.325     0.046  38.150     0.041  37.975  
   0.036  37.800     0.032  37.625     0.028  37.450     0.024  37.275  
   0.021  37.100     0.017  36.925     0.014  36.750     0.012  36.575  
   0.009  36.400     0.007  36.225     0.005  36.050     0.004  35.875  
   0.002  35.700     0.001  35.525     0.001  35.350     0.000  35.175  
   0.000  35.000     0.000  34.825     0.001  34.650     0.001  34.475  
   0.002  34.300     0.004  34.125     0.005  33.950     0.007  33.775  
   0.009  33.600     0.012  33.425     0.015  33.250     0.018  33.075  
   0.021  32.900     0.025  32.725     0.029  32.550     0.034  32.375  
   0.038  32.200     0.043  32.025     0.049  31.850     0.054  31.675  
   0.060  31.500     0.067  31.325     0.073  31.150     0.080  30.975  
   0.088  30.800     0.095  30.625     0.103  30.450     0.111  30.275  
   0.120  30.100     0.129  29.925     0.138  29.750     0.148  29.575  
   0.158  29.400     0.168  29.225     0.179  29.050     0.190  28.875  
   0.202  28.700     0.213  28.525     0.225  28.350     0.238  28.175  
   0.251  28.000     0.264  27.825     0.278  27.650     0.292  27.475  
   0.306  27.300     0.321  27.125     0.336  26.950     0.351  26.775  
   0.367  26.600     0.383  26.425     0.400  26.250     0.417  26.075  
   0.434  25.900     0.452  25.725     0.470  25.550     0.489  25.375  
   0.508  25.200     0.527  25.025     0.547  24.850     0.567  24.675  
   0.588  24.500     0.609  24.325     0.630  24.150     0.652  23.975  
   0.675  23.800     0.697  23.625     0.721  23.450     0.744  23.275  
   0.768  23.100     0.793  22.925     0.818  22.750     0.843  22.575  
   0.869  22.400     0.896  22.225     0.922  22.050     0.950  21.875  
   0.977  21.700     1.006  21.525     1.034  21.350     1.064  21.175  
   1.093  21.000     1.124  20.825     1.154  20.650     1.185  20.475  
   1.217  20.300     1.249  20.125     1.282  19.950     1.315  19.775  
   1.349  19.600     1.383  19.425     1.418  19.250     1.454  19.075  
   1.489  18.900     1.526  18.725     1.563  18.550     1.601  18.375  
   1.639  18.200     1.677  18.025     1.717  17.850     1.757  17.675  
   1.797  17.500     1.838  17.325     1.880  17.150     1.922  16.975  
   1.965  16.800     2.009  16.625     2.053  16.450     2.098  16.275  
   2.143  16.100     2.189  15.925     2.236  15.750     2.284  15.575  
   2.332  15.400     2.381  15.225     2.430  15.050     2.480  14.875  
   2.531  14.700     2.583  14.525     2.635  14.350     2.689  14.175  
   2.743  14.000     2.797  13.825     2.853  13.650     2.909  13.475  
   2.966  13.300     3.024  13.125     3.082  12.950     3.142  12.775  
   3.202  12.600     3.263  12.425     3.325  12.250     3.388  12.075  
   3.452  11.900     3.517  11.725     3.583  11.550     3.649  11.375  
   3.717  11.200     3.785  11.025     3.855  10.850     3.926  10.675  
   3.997  10.500     4.070  10.325     4.144  10.150     4.218   9.975  
   4.294   9.800     4.371   9.625     4.450   9.450     4.529   9.275  
   4.610   9.100     4.692   8.925     4.775   8.750     4.859   8.575  
   4.945   8.400     5.032   8.225     5.121   8.050     5.210   7.875  
   5.302   7.700     5.395   7.525     5.489   7.350     5.585   7.175  
   5.683   7.000     5.782   6.825     5.883   6.650     5.986   6.475  
   6.090   6.300     6.197   6.125     6.305   5.950     6.416   5.775  
   6.529   5.600     6.643   5.425     6.760   5.250     6.880   5.075  
   7.002   4.900     7.126   4.725     7.253   4.550     7.383   4.375  
   7.516   4.200     7.651   4.025     7.790   3.850     7.933   3.675  
   8.079   3.500     8.229   3.325     8.383   3.150     8.541   2.975  
   8.704   2.800     8.872   2.625     9.045   2.450     9.224   2.275  
   9.410   2.100     9.603   1.925     9.804   1.750    10.014   1.575  
  10.234   1.400    10.467   1.225    10.713   1.050    10.977   0.875  
  11.262   0.700    11.577   0.525    11.935   0.350    12.369   0.175  
  13.125   0.000    ] scurve
  13.125   0.000  moveto
 0.700 disk
  1.353  70.000  moveto
[  1.353  70.000     1.341  69.825     1.328  69.650     1.316  69.475  
   1.304  69.300     1.292  69.125     1.281  68.950     1.269  68.775  
   1.257  68.600     1.245  68.425     1.233  68.250     1.222  68.075  
   1.210  67.900     1.199  67.725     1.187  67.550     1.176  67.375  
   1.164  67.200     1.153  67.025     1.141  66.850     1.130  66.675  
   1.119  66.500     1.108  66.325     1.096  66.150     1.085  65.975  
   1.074  65.800     1.063  65.625     1.052  65.450     1.041  65.275  
   1.030  65.100     1.019  64.925     1.009  64.750     0.998  64.575  
   0.987  64.400     0.977  64.225     0.966  64.050     0.955  63.875  
   0.945  63.700     0.934  63.525     0.924  63.350     0.914  63.175  
   0.903  63.000     0.893  62.825     0.883  62.650     0.873  62.475  
   0.863  62.300     0.852  62.125     0.842  61.950     0.832  61.775  
   0.823  61.600     0.813  61.425     0.803  61.250     0.793  61.075  
   0.783  60.900     0.774  60.725     0.764  60.550     0.755  60.375  
   0.745  60.200     0.736  60.025     0.726  59.850     0.717  59.675  
   0.707  59.500     0.698  59.325     0.689  59.150     0.680  58.975  
   0.671  58.800     0.662  58.625     0.653  58.450     0.644  58.275  
   0.635  58.100     0.626  57.925     0.617  57.750     0.608  57.575  
   0.600  57.400     0.591  57.225     0.582  57.050     0.574  56.875  
   0.565  56.700     0.557  56.525     0.549  56.350     0.540  56.175  
   0.532  56.000     0.524  55.825     0.516  55.650     0.508  55.475  
   0.500  55.300     0.492  55.125     0.484  54.950     0.476  54.775  
   0.468  54.600     0.460  54.425     0.453  54.250     0.445  54.075  
   0.437  53.900     0.430  53.725     0.422  53.550     0.415  53.375  
   0.408  53.200     0.400  53.025     0.393  52.850     0.386  52.675  
   0.379  52.500     0.372  52.325     0.365  52.150     0.358  51.975  
   0.351  51.800     0.344  51.625     0.337  51.450     0.331  51.275  
   0.324  51.100     0.317  50.925     0.311  50.750     0.304  50.575  
   0.298  50.400     0.291  50.225     0.285  50.050     0.279  49.875  
   0.273  49.700     0.267  49.525     0.261  49.350     0.255  49.175  
   0.249  49.000     0.243  48.825     0.237  48.650     0.231  48.475  
   0.226  48.300     0.220  48.125     0.215  47.950     0.209  47.775  
   0.204  47.600     0.198  47.425     0.193  47.250     0.188  47.075  
   0.183  46.900     0.178  46.725     0.173  46.550     0.168  46.375  
   0.163  46.200     0.158  46.025     0.153  45.850     0.148  45.675  
   0.144  45.500     0.139  45.325     0.135  45.150     0.130  44.975  
   0.126  44.800     0.122  44.625     0.117  44.450     0.113  44.275  
   0.109  44.100     0.105  43.925     0.101  43.750     0.097  43.575  
   0.094  43.400     0.090  43.225     0.086  43.050     0.083  42.875  
   0.079  42.700     0.076  42.525     0.072  42.350     0.069  42.175  
   0.066  42.000     0.063  41.825     0.060  41.650     0.057  41.475  
   0.054  41.300     0.051  41.125     0.048  40.950     0.045  40.775  
   0.043  40.600     0.040  40.425     0.038  40.250     0.035  40.075  
   0.033  39.900     0.031  39.725     0.028  39.550     0.026  39.375  
   0.024  39.200     0.022  39.025     0.020  38.850     0.019  38.675  
   0.017  38.500     0.015  38.325     0.014  38.150     0.012  37.975  
   0.011  37.800     0.010  37.625     0.008  37.450     0.007  37.275  
   0.006  37.100     0.005  36.925     0.004  36.750     0.003  36.575  
   0.003  36.400     0.002  36.225     0.002  36.050     0.001  35.875  
   0.001  35.700     0.000  35.525     0.000  35.350     0.000  35.175  
  -0.000  35.000     0.000  34.825     0.000  34.650     0.000  34.475  
   0.001  34.300     0.001  34.125     0.002  33.950     0.002  33.775  
   0.003  33.600     0.004  33.425     0.004  33.250     0.005  33.075  
   0.006  32.900     0.008  32.725     0.009  32.550     0.010  32.375  
   0.012  32.200     0.013  32.025     0.015  31.850     0.016  31.675  
   0.018  31.500     0.020  31.325     0.022  31.150     0.024  30.975  
   0.026  30.800     0.029  30.625     0.031  30.450     0.033  30.275  
   0.036  30.100     0.039  29.925     0.042  29.750     0.044  29.575  
   0.047  29.400     0.051  29.225     0.054  29.050     0.057  28.875  
   0.060  28.700     0.064  28.525     0.068  28.350     0.071  28.175  
   0.075  28.000     0.079  27.825     0.083  27.650     0.087  27.475  
   0.092  27.300     0.096  27.125     0.101  26.950     0.105  26.775  
   0.110  26.600     0.115  26.425     0.120  26.250     0.125  26.075  
   0.130  25.900     0.135  25.725     0.141  25.550     0.146  25.375  
   0.152  25.200     0.158  25.025     0.164  24.850     0.170  24.675  
   0.176  24.500     0.182  24.325     0.189  24.150     0.195  23.975  
   0.202  23.800     0.209  23.625     0.216  23.450     0.223  23.275  
   0.230  23.100     0.238  22.925     0.245  22.750     0.253  22.575  
   0.260  22.400     0.268  22.225     0.276  22.050     0.284  21.875  
   0.293  21.700     0.301  21.525     0.310  21.350     0.318  21.175  
   0.327  21.000     0.336  20.825     0.346  20.650     0.355  20.475  
   0.364  20.300     0.374  20.125     0.384  19.950     0.394  19.775  
   0.404  19.600     0.414  19.425     0.424  19.250     0.435  19.075  
   0.446  18.900     0.456  18.725     0.467  18.550     0.479  18.375  
   0.490  18.200     0.502  18.025     0.513  17.850     0.525  17.675  
   0.537  17.500     0.549  17.325     0.562  17.150     0.574  16.975  
   0.587  16.800     0.600  16.625     0.613  16.450     0.627  16.275  
   0.640  16.100     0.654  15.925     0.668  15.750     0.682  15.575  
   0.696  15.400     0.711  15.225     0.725  15.050     0.740  14.875  
   0.755  14.700     0.770  14.525     0.786  14.350     0.802  14.175  
   0.818  14.000     0.834  13.825     0.850  13.650     0.867  13.475  
   0.884  13.300     0.901  13.125     0.918  12.950     0.936  12.775  
   0.953  12.600     0.971  12.425     0.990  12.250     1.008  12.075  
   1.027  11.900     1.046  11.725     1.065  11.550     1.085  11.375  
   1.105  11.200     1.125  11.025     1.145  10.850     1.166  10.675  
   1.187  10.500     1.208  10.325     1.230  10.150     1.252   9.975  
   1.274   9.800     1.296   9.625     1.319   9.450     1.342   9.275  
   1.366   9.100     1.390   8.925     1.414   8.750     1.438   8.575  
   1.463   8.400     1.488   8.225     1.514   8.050     1.540   7.875  
   1.566   7.700     1.593   7.525     1.620   7.350     1.648   7.175  
   1.676   7.000     1.705   6.825     1.734   6.650     1.763   6.475  
   1.793   6.300     1.823   6.125     1.854   5.950     1.886   5.775  
   1.918   5.600     1.950   5.425     1.983   5.250     2.017   5.075  
   2.051   4.900     2.086   4.725     2.122   4.550     2.158   4.375  
   2.195   4.200     2.233   4.025     2.272   3.850     2.311   3.675  
   2.351   3.500     2.392   3.325     2.434   3.150     2.477   2.975  
   2.521   2.800     2.567   2.625     2.613   2.450     2.661   2.275  
   2.710   2.100     2.761   1.925     2.813   1.750     2.867   1.575  
   2.923   1.400     2.982   1.225     3.043   1.050     3.107   0.875  
   3.175   0.700     3.248   0.525     3.328   0.350     3.417   0.175  
   3.535   0.000    ] scurve
   3.535   0.000  moveto
 0.700 disk
grestore}
\LabelTeX    2.000 34.000 $C_0\pmatrix{0\cr r_0\cr}$\ELTX
\LabelTeX   64.000  6.000 $M(t)\pmatrix{vt\cr 0\cr}$\ELTX
\LabelTeX   88.000 -3.500 $x$\ELTX
\LabelTeX   -4.000 71.500 $y$\ELTX
\LabelTeX  -15.700 -5.500 $\lambda=V/v=10$\ELTX
\LabelTeX   11.000 -5.500 $3,0$\ELTX
\LabelTeX   21.000 -5.500 $2,0$\ELTX
\LabelTeX   39.000 -5.500 $1,5$\ELTX
\LabelTeX   74.000 -5.500 $1,25$\ELTX
\EndFig
\vskip15pt

\state Remarque -- | Les \'equations ont encore un sens lorsque $t<0$.
On a dans ce cas $\alpha\in{}]\pi/2,\pi[\,$, $\theta\in{}]1,+\infty[\,$, 
le chien se trouve dans
le quadrant $x>0,\, y>r_0$ et se dirige vers le ma\^{\i}tre qui
parcourt de son c\^ot\'e la demi-droite $x<0$.

\section 4. \'Equations diff\'erentielles du second ordre|

\subsection 4.1. Remarques g\'en\'erales|

\noindent On consid\`ere une \'equation diff\'erentielle
$$
y''=f(x,y,y')\leqno{\rm (E)}
$$
o\`u $f:U\rightarrow\bR$, $U\subset \bR^3$, est une application continue
localement lipschitzienne en ses deuxi\`eme et troisi\`eme variables.

La solution g\'en\'erale $y$ d\'efinie au voisinage d'un point $x_0$ d\'epend
alors de deux param\`etres $\lambda,\mu\in\bR$ qui apparaissent le plus
souvent comme des constantes d'int\'egration~:
$$
y(x)=\varphi(x,\lambda,\mu).
$$
Le th\'eor\`eme de Cauchy-Lipschitz montre qu'on peut choisir $y_0=y(x_0)$,
$y_1=y'(x_0)$ comme param\`etres.

Il existe tr\`es peu de cas o\`u on sait r\'esoudre explicitement une \'equation du second
ordre~: m\^eme les \'equations lin\'eaires du second ordre sans second membre ne se
r\'esolvent pas explicitement en g\'en\'eral.

\subsection 4.2. \'Equations incompl\`etes du second ordre|

\noindent {\tenpointbf a) \'Equations du type~ (E)~:\quad $y''=f(x,y')$}
\medskip
Si on consid\`ere la nouvelle fonction inconnue $v=y'$, (E) se
ram\`ene \`a l'\'equation du premier ordre
$$
v'=f(x,v).
$$
La solution g\'en\'erale de cette derni\`ere sera de la forme $v(x,\lambda)$,
$\lambda\in\bR$, et on obtient donc
$$
y(x)=\int^x_{x_0}v(t,\lambda)dt+\mu,\quad \mu\in\bR.
$$

\bigskip
\noindent {\tenpointbf b) \'Equations du type~ (E)~:\quad $y''=f(y,y')$}
\medskip
La m\'ethode consiste \`a prendre $y$ comme nouvelle variable et
$v=y'$ comme variable fonction inconnue (en la variable $y$).

\medskip
\noindent$\bullet$ Il peut y avoir des solutions constantes $y(x)=y_0$, 
auquel cas $y$ ne peut \^etre choisi comme variable. On a donc des solutions
singuli\`eres
$$
y(x)=y_j,\quad{\rm avec}\quad f(y_j,0)=0.
$$
$\bullet$ Cas g\'en\'eral
$$
y''=\frac{dy'}{dx}=\frac{dy}{dx}\cdot\frac{dy'}{dy}=v\ \frac{dv}{dy}
$$
L'\'equation se ram\`ene alors \`a l'\'equation du premier ordre
$$
v\ \frac{dv}{dy}=f(y,v).
$$
La r\'esolution de cette derni\`ere donne une solution g\'en\'erale 
$v(y,\lambda)$, $\lambda\in\bR$. On doit ensuite r\'esoudre
$$
y'=v(y,\lambda)\Leftrightarrow\frac{dy}{v(y,\lambda)}=dx,
$$
d'o\`u la solution g\'en\'erale
$$
\int\frac{dy}{v(y,\lambda)}=x+\mu,\qquad \mu\in\bR.
$$

\bigskip
\noindent {\tenpointbf c) \'Equations du type~ (E)~:\quad $y''=f(y)$}
\medskip
C'est un cas particulier du cas b) pr\'ec\'edent, mais on peut ici pr\'eciser
davantage la m\'ethode de r\'esolution. On a en effet
$$
y'y''=f(y)y',
$$
et en int\'egrant il vient $\dfrac{1}{2}\,y'^2=\varphi(y)+\lambda$, 
$\lambda\in\bR$, o\`u $\varphi$ est une primitive de $f$. On obtient donc
$$
\eqalign{y'=&\pm \sqrt{2(\varphi(y)+\lambda)},\cr
&\pm\frac{dy}{\sqrt{2(\varphi(y)+\lambda)}}=dx,\cr
&\pm\int^y_{y_0}\frac{du}{\sqrt{2(\varphi(u)+\lambda)}}=x+\mu,\quad \mu\in\bR.\cr}
$$

\bigskip
\state Interpr\'etation physique -- |  On \'etudie la loi du mouvement
d'un point mat\'eriel $M$ de masse $m$ astreint \`a se d\'eplacer sur
une courbe $(C)$. On suppose que la composante tangentielle
$\overrightarrow{F_T}$ de la force $\overrightarrow F$ qui s'exerce sur
$M$ ne d\'epend que de la position de $M$, rep\'er\'ee par son abscisse
curviligne $y$ sur $(C)$.

\InsertFig 36.000 42.000 {
1 mm unit
  0.000   0.000 moveto 
[   0.000   0.000    2.000  10.000   14.000  19.000   26.000  22.000  
 41.000  22.000   54.000  18.000  ] curve 
stroke 
 14.000  19.000 moveto 
  0.700 disk 
  0.340 setlinewidth 
 20.850  21.000 vector 
 15.232 -69.000 vector 
 25.800 -14.859 vector 
 47.000  25.000 moveto 
  0.113 setlinewidth 
  8.382 -17.354 vector 
[   2.000   1.000 ] 0 setdash 
 33.700  26.600 moveto 
 39.200  12.500 lineto 
stroke 
 20.000   4.500 moveto
 38.500  12.000 lineto 
stroke}
\LabelTeX   -5.500   4.900 $(C)$\ELTX
\LabelTeX   10.400  20.900 $M$\ELTX
\LabelTeX   34.000  28.000 $\overrightarrow{F_T}$\ELTX
\LabelTeX   40.000  11.000 $\overrightarrow{F}$\ELTX
\LabelTeX   19.000   0.000 $\overrightarrow{F_N}$\ELTX
\LabelTeX   50.500  26.000 $y$\ELTX
\EndFig
\vskip8mm

\noindent Par hypoth\`ese, il existe une fonction $f$ telle que $F_T=f(y)$. 
Le principe fondamental de la dynamique donne
$$
F_T=m\gamma_T=m\ \frac{d^2y}{dt^2},
$$
d'o\`u
$$
my''=f(y)
\leqno{\rm(E)}
$$
avec $y''=d^2y/dt^2$. On en d\'eduit $m y'y''-f(y)y'=0$, donc $\dfrac{1}{2}\
my'^2-\varphi(y)=\lambda$, o\`u $\varphi$ est une primitive de $f$. La quantit\'e
$\dfrac{1}{2}\ my'^2=E_{\rm c}$ est ``l'\'energie cin\'etique'' de la particule
tandis que
$$
-\varphi(y)=-\int f(y)dy=-\int\overrightarrow{F_T}(M)\cdot d\overrightarrow
M=-\int\overrightarrow F(M)\cdot d\overrightarrow M 
$$
est ``l'\'energie potentielle'' $E_{\rm p}$. L'\'energie totale
$$
E_{\rm t}=E_{\rm c}+E_{\rm p}=\frac{1}{2}\ my'^2-\varphi(y)
$$
est constante quel que soit le mouvement du point $M$. On dit que
$U(y,y')= \frac{1}{2}\ my'^2-\varphi(y)$ est une ``int\'egrale
premi\`ere'' de (E) (dans le sens que c'est une relation
diff\'erentielle obtenue \`a l'aide d'une premi\`ere int\'egration de
l'\'equation du second ordre, une deuxi\`eme int\'egration restant
n\'ecessaire pour \'etablir la loi du mouvement). Si $E_{\rm t}$ d\'esigne
l'\'energie totale, la loi du mouvement est donn\'ee par
$$
t-t_0=\pm\sqrt{m/2}\int_{y_0}^y{du\over\sqrt{E_{\rm t}+\varphi(u)}}
$$
au voisinage de tout donn\'ee initiale $(t_0,y_0,y_0')$ telle que 
${1\over 2}my_0^{\prime2}=E_{\rm t}+\varphi(y_0)>0$.

\bigskip
\state Compl\'ement -- |  Supposons que la fonction $f:\bR\to\bR$ soit
de classe $C^1$, strictement d\'ecroissante et telle que $f(0)=0$~; 
on a donc en particulier $f(y)>0$ pour $y<0$ et $f(y)<0$ pour $y<0$
(physiquement, ceci signifie que la force est une ``force de rappel''
vers la position neutre $y=0$, dont l'intensit\'e s'accro\^{\i}t avec
la distance \`a la position neutre). Alors les solutions maximales
$t\mapsto y(t)$ sont {\it p\'eriodiques}. Pour le voir, posons par exemple
$\varphi(u)=\int_0^uf(y)dy$ et observons que $\varphi$ est une fonction
concave n\'egative ou nulle, passant par un maximum en $\varphi(0)=0$.
Comme ${1\over 2}my^{\prime 2}-\varphi(y)=E_{\rm t}$ avec $y^{\prime 2}
\ge 0$ et $-\varphi(y)>0$ si $y\ne 0$, les solutions non triviales
n'existent que pour une valeur $E_{\rm t}>0$ de l'\'energie totale, et
elles v\'erifient $|y'|\le\sqrt{2E_{\rm t}/m}$ et $-\varphi(y)\le
E_{\rm t}$. De plus, si $a<0$, $b>0$ sont les uniques r\'eels
n\'egatif et positif tels que $\varphi(a)=\varphi(b)=-E_{\rm t}$, on a
$a\le y(t)\le b$ pour tout $t$. Ceci implique d\'ej\`a que les
solutions maximales sont d\'efinies sur $\bR$ tout entier [si par
exemple une solution maximale n'\'etait d\'efinie que sur un
intervalle ouvert $]t_1,t_2[$, le crit\`ere de Cauchy uniforme
montrerait que $y$ se prolonge par continuit\'e \`a droite en $t_1$ 
et \`a gauche en $t_2$, puisque $y$ est lipschitzienne de
rapport${}\leq\sqrt{2E_{\rm t}/m}$, et de m\^eme $y'$ et $y''$ se
prolongeraient gr\^ace \`a la relation $y''={1\over m}f(y)$~;
l'existence de solutions locales au voisinage de $t_1$ et~$t_2$
contredirait alors la maximalit\'e de~$y$]. La relation
${1\over 2}my^{\prime 2}-\varphi(y)=E_{\rm t}$ montre que $y'\ne 0$ 
lorsque $a<y<b$ (car on a alors $-\varphi(y)<E_{\rm t}$). Par continuit\'e,
la fonction $y'$ est donc de signe constant sur tout intervalle
de temps o\`u $a<y<b$. La solution explicite donn\'ee plus haut montre 
que $y$ est alternativement croissant de $a$ \`a $b$ puis d\'ecroissant 
de $b$ \`a $a$, avec demi-p\'eriode
$$
{T\over 2}=\sqrt{m/2}\int_a^b{du\over\sqrt{E_{\rm t}+\varphi(u)}},
$$
et, en choisissant $t_0$ tel que $y(t_0)=\min y=a$, on a les relations
$$
\matrix{
&\kern-10pt t=t_0+\sqrt{m/2}\displaystyle
\int_a^y{du\over\sqrt{E_{\rm t}+\varphi(u)}}+nT,\hfill
&t\in[t_0+nT,t_0+nT+T/2],\hfill\cr
&\kern-10pt t=t_0-\sqrt{m/2}\displaystyle
\int_a^y{du\over\sqrt{E_{\rm t}+\varphi(u)}}+(n+1)T,\hfill
&t\in[t_0+nT+T/2,t_0+(n+1)T].\hfill\cr}
$$
On observera que l'int\'egrale donnant la p\'eriode $T$ est convergente,
car on a $\varphi'(a)=f(a)>0$, $\varphi'(b)=f(b)<0$, de sorte que
$E_{\rm t}+\varphi(u)\sim f(a)(u-a)$ au voisinage de~$a$, et de m\^eme
au voisinage de~$b$.

\bigskip
\state Exemple -- |  Mouvement d'un pendule simple de masse $m$ suspendu
\`a un fil
de longueur $l$.

\InsertFig 36.000 28.000 {
1 mm unit
  0.000   0.000 moveto 
 18.000  24.000  30.000 233.130 336.371 arc stroke 
 18.000  24.000 moveto 
  0.700 disk 
 36.000   0.000 moveto 
  1.000 disk 
 18.000  24.000 lineto 
stroke 
 24.000  -9.000 moveto 
 48.000   9.000 lineto 
stroke 
  0.340 setlinewidth 
 36.000   0.000 moveto 
 15.000 -90.000 vector 
  9.000 216.870 vector 
 10.548 185.440 vector 
 17.500 126.870 vector
  0.113 setlinewidth 
[   2.000   1.000 ] 0 setdash
 18.000  -6.000 moveto 
 18.000  24.000 lineto 
stroke 
 25.500  -1.000 moveto 
 25.500  14.000 lineto 
stroke 
 25.500  -1.000 moveto 
 36.000 -15.000 lineto 
stroke 
 36.000  -9.000 moveto 
[ ] 0 setdash 
 36.000 -15.000   6.000  90.000 129.806 arc stroke 
 18.000  16.000 moveto 
 18.000  24.000   8.000 -90.000 -53.130 arc stroke 
 22.800  17.600 moveto 
  0.000  29.000 vector 
 45.555  12.138 moveto 
  0.000  66.371 vector}
\LabelTeX   15.600    8.000 $l$\ELTX
\LabelTeX   19.500   13.000 $\theta$\ELTX
\LabelTeX   38.000   -2.000 $m$\ELTX
\LabelTeX   42.000   11.500 $y$\ELTX
\LabelTeX   28.500   13.000 $\overrightarrow{T}$\ELTX
\LabelTeX   22.000   -3.500 $\overrightarrow{F}$\ELTX
\LabelTeX   26.000  -11.500 $\overrightarrow{F_{\hbox{\rlap{$T$}}}}$\ELTX
\LabelTeX   37.500  -18.000 $\overrightarrow{P}=m\overrightarrow{g}$\ELTX
\LabelTeX   32.100   -8.200 $\theta$\ELTX
\EndFig
\vskip10mm

\noindent On a ici $y=l\theta$ et $F_T=P\,\sin\,\theta=-mg\,\sin\,\theta$,
d'o\`u
$$
\eqalign{
&ml\theta''=-mg\,\sin\,\theta,\cr
&\theta''=-\dfrac{g}{l}\ \sin\,\theta.\cr
}
$$
L'\'energie totale est
$$
E_{\rm t}=E_{\rm c}+E_{\rm p}=\frac{1}{2}\,my'^2-mgl\,\cos\,\theta=
\frac{1}{2}\,ml^2\theta'^2-mgl\,\cos\,\theta. 
$$
Les solutions $t\mapsto\theta(t)$ v\'erifient
$$
\eqalign{
&\theta'^2-\frac{2g}{l}\ \cos\,\theta=\lambda={2E_{\rm t}\over ml^2},
\qquad\lambda\in\bR,\cr
&\pm\int^{\theta}_0\frac{d\varphi}{\sqrt{\lambda+\dfrac{2g}{l}\,\cos\,\varphi}}
=t-t_0,\qquad t_0\in\bR.\cr}
$$
L'int\'egrale ne se calcule pas explicitement, sauf si
$\lambda=\dfrac{2g}{l}$, auquel cas
$$
t-t_0=\pm\sqrt{\dfrac{l}{g}}\int^{\theta}_0\frac{d\varphi}{2\cos\varphi/2}
=\pm\sqrt{\dfrac{l}{g}}\,\ln\tan\Big(\dfrac{\theta}{4}+\dfrac{\pi}{4}\Big),
$$
et le pendule atteint la position verticale haute $\theta=\pm\pi$ en un temps 
infini. Dans les autres cas, les solutions
maximales sont p\'eriodiques. Si $\lambda<2g/l$, l'\'equation diff\'erentielle
implique $\cos\theta\geq-\lambda l/2g>-1$ et l'amplitude angulaire
est donc major\'ee en valeur absolue par une amplitude maximale 
$\theta_m\in{}]0,\pi[$ 
telle que $\cos\theta_m=-\lambda l/2g$~; dans ce cas le mouvement est 
oscillatoire autour de la position d'\'equilibre $\theta=0$ (ceci correspond 
\`a la situation \'etudi\'ee dans la remarque, avec une fonction 
$f(\theta)=-\sin\theta$ strictement d\'ecroissante sur $[-\theta_m,\theta_m])$.
La demi-p\'eriode est donn\'ee par
$$
{T\over 2}=\int_{-\theta_m}^{\theta_m}
\frac{d\varphi}{\sqrt{\lambda+\dfrac{2g}{l}\,\cos\,\varphi}}=
2\int_0^{\theta_m}\frac{d\varphi}{\sqrt{\lambda+\dfrac{2g}{l}\,\cos\,\varphi}}.
$$
Si $\lambda>2g/l$, on n'est plus dans la situation de la remarque,
mais on a cependant encore un mouvement p\'eriodique de demi-p\'eriode
$$
{T\over 2}=\int_0^\pi\frac{d\varphi}
{\sqrt{\lambda+\dfrac{2g}{l}\,\cos\,\varphi}},
$$
le pendule effectuant des rotations compl\`etes sans jamais changer de
sens de rotation.

En physique, on s'int\'eresse
g\'en\'eralement aux oscillations de faible amplitude du pendule. Ceci
permet de faire l'approximation usuelle $\sin\,\theta\simeq\theta$ et
on obtient alors les solutions approch\'ees classiques
$\theta=\theta_m\cos\,\omega(t-t_0)$ avec $\omega=\sqrt{g/l}$. Nous
reviendrons sur cette question au paragraphe 2.4 du chapitre XI, et nous 
indiquerons en particulier une m\'ethode permettant d'\'evaluer l'erreur
commise.

\subsection 4.3. \'Equations lin\'eaires homog\`enes du second ordre|

\noindent La th\'eorie g\'en\'erale des \'equations et syst\`emes 
diff\'erentiels lin\'eaires sera faite au chapitre suivant. Indiquons
un cas o\`u l'on peut se ramener \`a un calcul de primitives. Soit
$$
a(x)y''+b(x)y'+c(x)y=0.\leqno{\rm (E)}
$$
Supposons qu'on connaisse une solution particuli\`ere $y_{(1)}$ de
(E). On peut alors chercher la solution g\'en\'erale par la m\'ethode
de variation des constantes~:
$$
y(x)=\lambda(x)y_{(1)}(x).
$$
Il vient
$$
\displaylines{
a(x)\Big(\lambda''y_{(1)}+2\lambda'y'_{(1)}+\lambda y''_{(1)}\Big)
+b(x)\Big(\lambda'y_{(1)}+\lambda y'_{(1)}\Big)+c(x)\lambda y_{(1)}=0,\cr
\noalign{\vskip6pt}
\lambda\Big(a(x)y''_{(1)}+b(x)y'_{(1)}+c(x)y_{(1)}\Big)
+\lambda'\Big(2a(x)y'_{(1)}+b(x)y_{(1)}\Big)+\lambda''a(x)y_{(1)}=0,\cr
\lambda'\Big(2a(x)y'_{(1)}+b(x)y_{(1)}\Big)+\lambda''a(x)y_{(1)}=0.\cr
}
$$
La fonction $\mu=\lambda'$ est donc solution d'une \'equation diff\'erentielle lin\'eaire
du premier ordre, qui se peut se r\'ecrire
$$
\frac{\mu'}{\mu}=\frac{\lambda''}{\lambda'}=-2\
\frac{y'_{(1)}}{y_{(1)}}-\frac{b(x)}{a(x)}. 
$$
La solution g\'en\'erale est donn\'ee par $\mu=\alpha\mu_{(1)}$, $\alpha\in\bR$, d'o\`u
$\lambda=\alpha\lambda_{(1)}+\beta$, $\beta\in\bR$, o\`u $\lambda_{(1)}$ est une primitive
de $\mu_{(1)}$. La solution g\'en\'erale de (E) est donc~:
$$
y(x)=\alpha\lambda_{(1)}(x)y_{(1)}(x)+\beta y_{(1)}(x),\quad(\alpha,\beta)\in\bR^2.
$$
Les solutions forment un espace vectoriel de dimension 2.

\bigskip
\majedeb
\state Exercice -- |  {\it R\'esoudre $x^2(1-x^2)y''+x^3y'-2y=0$ en observant que
$y_{(1)}(x)=x^2$ est solution.} 
\par\majefin\par

\subsection 4.4.* \'Equations diff\'erentielles issues de probl\`emes 
va\-ria\-tionnels|

\noindent Les probl\`emes variationnels conduisent tr\`es souvent \`a
la r\'esolution d'\'equations diff\'erentielles du second ordre. Avant
de donner un exemple, nous allons r\'esoudre un probl\`eme variationnel
g\'en\'eral dans une situation simple. On consid\`ere un op\'erateur
fonctionnel (c'est-\`a-dire une fonction dont la variable est une
fonction)
$$
\eqalign{
\varphi:C^2([a,b])&\rightarrow\bR\cr
u&\mapsto\varphi(u)=\int^b_aF(x,u(x),u'(x))dx,\cr
}
$$
o\`u $F:[a,b]\times\bR\times\bR\rightarrow\bR$, $(x,y,z)\mapsto
F(x,y,z)$ est une application de classe $C^2$. Le probl\`eme typique du
calcul des variations est de rechercher les extrema de $\varphi(u)$
lorsque $u$ d\'ecrit $C^2([a,b])$ avec la \og$\,$contrainte aux bornes$\,$\fg\
suivante~: les valeurs aux bornes de l'intervalle $u(a)=u_1$, $u(b)=u_2$
sont fix\'ees.

Soit $h\in C^2([a,b])$ avec $h(a)=h(b)=0$. Pour tout $t\in\bR$, la
fonction $u+th$ v\'erifie la m\^eme contrainte aux bornes que la
fonction $u$. Si $u$ est un extremum de $\varphi$ sous les conditions
pr\'ecis\'ees plus haut, alors $t=0$ est un extremum de la fonction
d'une variable r\'eelle
$$
\psi_h(t)=\varphi(u+th)=\int^b_aF(x,u(x)+th(x),u'(x)+th'(x))dx.
$$
On doit donc avoir $\psi'(0)=0$, et ceci quel que soit la fonction $h\in C^2([a,b])$
v\'erifiant $h(a)=h(b)=0$. D'apr\`es le th\'eor\`eme de d\'erivation sous le signe somme
il vient
$$
\psi'_h(0)=\int^b_a\bigl(h(x)F'_y(x,u,u')+h'(x)F'_z(x,u,u')\bigl)dx.
$$
En int\'egrant par parties le terme en $h'(x)$ on obtient
$$
\psi'_h(0)=\int^b_a h(x)\Big(F'_y(x,u,u')-\frac{d}{dx}\ F'_z(x,u,u')\Big)dx.
$$
Par densit\'e de l'ensemble des fonctions $h$ consid\'er\'ees dans l'espace $L^1([a,b])$
des fonctions int\'egrables sur $[a,b]$, on aura donc $\psi'_h(0)=0$ pour tout $h$ si et
seulement si $u$ satisfait l'\'equation diff\'erentielle
$$
F'_y(x,u,u')-\frac{d}{dx}\ \Big(F'_z(x,u,u')\Big)=0,\leqno{\rm (E)}
$$
ou encore~:
$$
F'_y(x,u,u')-F''_{xz}(x,u,u')-u'F''_{yz}(x,u,u')-u''F''_{zz}(x,u,u')=0.
$$
Cette \'equation diff\'erentielle du second ordre en $u$ est appel\'ee {\it \'equation
d'Euler-Lagrange} associ\'ee au probl\`eme variationnel d\'efini par l'op\'erateur $\varphi$.

\bigskip
\state Application \`a la cha\^{\i}nette${}^*\,$-- | On cherche \`a
d\'eterminer la courbe repr\'esentant la position \`a l'\'equilibre
d'un fil souple inextensible de masse lin\'eique $\mu=dm/ds$
constante, lorsque ce fil est suspendu par ses extr\'emit\'es en des
points situ\'es \`a la m\^eme hauteur (cette courbe est appel\'ee 
\og$\,$cha\^{\i}nette$\,$\fg). On admettra comme physiquement \'evident 
que la courbe cherch\'ee est sym\'etrique et situ\'ee dans le plan vertical
contenant les extr\'emit\'es. Soit $Oxy$ un rep\`ere orthonorm\'e de
ce plan tel que $Oy$ est la verticale orient\'e vers le haut et
passant par le point le plus bas de la courbe, les extr\'emit\'es
ayant pour coordonn\'ees $(\pm a,0)$. Soit enfin $s\in[-\ell/2,\ell/2]$
l'abscisse curviligne mesur\'ee le long du fil avec le point le plus
bas pris comme origine ($\ell$~d\'esigne la longueur du fil). La position
d'\'equilibre correspond \`a la position la plus basse possible du
centre de gravit\'e $G$. Par sym\'etrie, on a ($x$ \'etant choisi
comme variable)~:
$$
y_G=\frac{1}{m/2}\int^a_0 y\,dm=
\frac{1}{\mu \ell/2}\int^a_0 y\mu\, ds=\frac{2}{\ell}\int^a_0 y\,ds, 
$$
o\`u $m=\mu \ell$ est la masse du fil. Une int\'egration par parties donne
$$
\eqalign{
y_G&=\frac{2}{\ell}\ \big[ys\big]^a_0-\frac{2}{\ell}\int^a_0 s\,dy\cr
&=-\frac{2}{\ell}\int^a_0 s\,dy=-\frac{2}{\ell}\int^a_0 s\sqrt{s'^2-1}\,dx\cr
}
$$
avec $s'=ds/dx=\sqrt{dx^2+dy^2}/dx$. Le probl\`eme revient donc \`a d\'eterminer les
fonctions $s=s(x)$ r\'ealisant le maximum de l'op\'erateur
$$
\varphi(s)=\int^a_0 s\sqrt{s'^2-1}\,dx,
$$
avec les contraintes $s(0)=0$, $s(a)=\ell/2$. L'\'equation d'Euler-Lagrange appliqu\'ee \`a
$F(x,s,t)=s\sqrt{t^2-1}$ donne
$$
\sqrt{s'^2-1}-\frac{d}{dx}\
\Big(\frac{ss'}{\sqrt{s'^2-1}}\Big)=\sqrt{s'^2-1}-\frac{s'^2+ss''}{\sqrt{s'^2-1}}
+\frac{ss'^2s''}{(s'^2-1)^{3/2}}=0.
$$
Apr\`es multiplication par $(s'^2-1)^{3/2}$ on obtient l'\'equation
$$
(s'^2-1)^2-(s'^2-1)(s'^2+ss'')+ss'^2s''=1-s'^2+ss''=0.\leqno{\rm (E)}
$$
On r\'esout cette \'equation gr\^ace \`a la m\'ethode d\'ecrite au
paragraphe 4.2.b), consistant \`a choisir $s$ comme nouvelle variable
et $v=s'$ comme nouvelle fonction inconnue. Il vient successivement
$$
\eqalign{
s''&=\frac{ds'}{dx}=\frac{ds}{dx}\cdot\frac{ds'}{ds}=v\ \frac{dv}{ds},\cr
{\rm (E)}&\Rightarrow 1-v^2+sv\ \frac{dv}{ds}=0,\cr
\frac{ds}{s}&=\frac{vdv}{v^2-1}\Rightarrow\ln\,s=\frac{1}{2}\ \ln(v^2-1)+C,\cr
s&=\lambda\sqrt{v^2-1}=\lambda\sqrt{s'^2-1},\cr
s'&=\frac{ds}{dx}=\sqrt{1+\dfrac{s^2}{\lambda^2}}\Rightarrow
dx=\frac{ds}{\sqrt{1+s^2/\lambda^2}},\cr
x&=\lambda\,\Argsinh\dfrac{s}{\lambda}\Rightarrow
s=\lambda\,\sinh\,\dfrac{x}{\lambda},\cr
\frac{ds}{dx}&=\sqrt{1+\Big(\dfrac{dy}{dx}\Big)^2}={\rm ch}\
\frac{x}{\lambda}\Rightarrow\frac{dy}{dx}=\sinh\,\frac{x}{\lambda}.\cr   
}
$$
On en d\'eduit l'\'equation de la cha\^{\i}nette, en tenant compte
du fait que $y(a)=0$~:
$$
y=\lambda\Big(\cosh\,\frac{x}{\lambda}-\cosh\,\frac{a}{\lambda}\Big).
$$
Le param\`etre $\lambda$ se calcule \`a partir de la relation
$\lambda\,\sinh\,a/\lambda=\ell/2$, obtenue en \'egalant $s(a)=\ell/2$.

\bigskip
\state Remarque -- | Notre raisonnement n'est pas parfaitement
rigoureux dans la mesure o\`u $F(x,s,t)=s\sqrt{t^2-1}$ est de classe
$C^2$ seulement sur $\bR\times\bR\times\{|t|>1\}$, alors que $|s'|$
est $\geq 1$ mais prend la valeur 1 pour $x=0$ (on notera que
$ds/dx=1/\cos\,\theta$ o\`u $\theta$ est l'angle de la tangente \`a la
courbe avec l'axe $0x$). Supposons $s'(x)>1$ pour $x>0$, comme c'est
le cas pour la solution physique observ\'ee. Le raisonnement de
d\'erivation sous la signe somme et l'int\'egration par parties
appliqu\'es dans les consid\'erations g\'en\'erale du d\'ebut
fonctionnent encore pour $|t|$ petit si on suppose $h(x)=0$ sur un
voisinage de $0$ (et aussi bien s\^ur $h(a)=0$).

Ces fonctions $h$ sont encore denses dans $L^1([0,a])$, donc $s$ doit
effectivement satisfaire l'\'equation diff\'erentielle (E) sur
$]0,a]$.
\bigskip

\state Calcul de g\'eod\'esiques${}^{**}$ -- | Nous \'etudions ici une autre
application importante du calcul des variations, \`a savoir le calcul
des g\'eod\'esiques d'une surface (ou d'une vari\'et\'e de dimension
plus grande). Si nous avons une surface $S\subset\bR^3$ donn\'ee comme
un graphe $z=h(x,y)$ d'une fonction $h:\Omega\to\bR$ sur un ouvert
$\Omega\subset\bR^2$, l'\'el\'ement de
longueur infinit\'esimal de la surface $S$ est donn\'e pour tout
$(x,y)\in\Omega$ par
$$
\eqalign{
ds^2&=dx^2+dy^2+dz^2=dx^2+dy^2+(h'_xdx+h'_ydy)^2\cr
&=(1+h_x^{\prime\,2})dx^2+2h'_xh'_ydx\,dy+(1+h_y^{\prime\,2})dy^2.\cr}
$$
Plus g\'en\'eralement, une {\it m\'etrique riemannienne}
sur un ouvert $\Omega\subset\bR^m$ est une expression de l'\'el\'ement
de longueur infinit\'esimal par une forme quadratique d\'efinie positive,
d\'ependant du point $x\in\Omega$ consid\'er\'e~:
$$
ds^2=q(x,dx)=\sum_{1\le i,j\le m}a_{ij}(x)\,dx_i dx_j
$$
(avec une matrice sym\'etrique $(a_{ij}(x))$ d\'efinie positive).
On supposera en outre que les coefficients $a_{ij}(x)$ sont
suffisamment r\'eguliers, disons de classe $C^2$. \'Etant donn\'e une
courbe $\gamma:[a,b]\to \Omega$ de classe $C^1$, sa longueur
(riemannienne) est par d\'efinition
$$
\eqalign{
&ds=\Vert \gamma'(t)\,dt\Vert_q=\sqrt{q\big(\gamma(t),\gamma'(t)dt\big)}=
\sqrt{\sum_{1\le i,j\le m}a_{ij}(\gamma(t))\,\gamma'_i(t)\gamma'_j(t)}\,dt,\cr
&\lg(\gamma)=\int_a^bds=
\int_a^b\sqrt{\sum_{1\le i,j\le m}a_{ij}(\gamma(t))\,\gamma'_i(t)\gamma'_j(t)}
\,dt.\cr}
$$
Pour deux points $x,y\in\Omega$, la distance g\'eod\'esique
$d_q(x,y)$ est par d\'efinition $\inf_\gamma\lg(\gamma)$ pour tous les
chemins $\gamma:[a,b]\to\Omega$ de classe $C^1$ d'extr\^emit\'es
$\gamma(a)=x$, $\gamma(b)=y$. Si un chemin r\'ealise l'infimum, on dit
qu'il s'agit d'une g\'eod\'esique de la m\'etrique riemannienne (on
notera qu'en g\'en\'eral un tel chemin n'existe pas n\'ecessairement,
et s'il existe il peut ne pas \^etre unique).  Un probl\`eme
fondamental est de d\'eterminer l'\'equation des g\'eod\'esiques afin
entre autres de calculer la distance g\'eod\'esique. Pour cela il est
commode d'introduire \og$\,$l'\'energie d'un chemin$\,$\fg\ qui est par
d\'efinition
$$
E(\gamma)=\int_a^b\Vert\gamma'(t)\Vert_q^2\,dt
=\int_a^b\sum_{1\le i,j\le m}a_{ij}(\gamma(t))\,\gamma'_i(t)\gamma'_j(t)
\,dt.
$$
L'in\'egalit\'e de Cauchy-Schwarz donne
$$
\Big(\int_a^b\Vert\gamma'(t)\Vert_q\,dt\Big)^2
=\Big(\int_a^b1\cdot \Vert\gamma'(t)\Vert_q\,dt\Big)^2\le
(b-a)\int_a^b\Vert\gamma'(t)\Vert_q^2\,dt
$$
soit $\lg(\gamma)\le\big((b-a)E(\gamma)\big)^{1/2}$, avec \'egalit\'e
si et seulement si 
$$
{ds\over dt}=\Vert\gamma'(t)\Vert_q=\hbox{Cte},
$$
condition qui peut toujours \^etre r\'ealis\'ee en reparam\'etrisant
le chemin $\gamma$ par son abscisse curviligne~$s$. Il en r\'esulte que
les chemins qui minimisent l'\'energie sont exactement les g\'eod\'esiques 
param\'etr\'ees par l'abscisse curviligne (\`a un facteur constant pr\`es).
Or la fonctionnelle d'\'energie $\gamma\mapsto E(\gamma)$
admet pour diff\'erentielle
$$
\eqalign{
E'(\gamma)\cdot h&=\int_a^b\Big(\sum_{i,j,k}{\partial a_{ij}\over
\partial x_k}(\gamma(t))\,\gamma'_i(t)\gamma'_j(t)h_k(t)
+2\sum_{i,j}a_{ij}(\gamma(t))\,\gamma'_i(t)h'_j(t)\Big)\,dt\cr
&=\int_a^b\sum_k h_k(t)\Big(\sum_{i,j}
{\partial a_{ij}\over\partial x_k}(\gamma(t))\,
\gamma'_i(t)\gamma'_j(t)-2{d\over dt}\sum_ia_{ik}(\gamma(t))\,\gamma'_i(t)
\Big)\,dt\cr}
$$
apr\`es int\'egration par parties (on suppose bien s\^ur $h_j(a)=h_j(b)=0$).
Il en r\'esulte que le coefficient de chaque terme $h_k(t)$ doit \^etre
identiquement nul.  En multipliant par $-1/2$ et en d\'eveloppant 
la d\'eriv\'ee $d/dt$,
on obtient le syst\`eme d'\'equations d'Euler-Lagrange caract\'erisant
les g\'eod\'esiques~:
$$
\sum_ia_{ik}(\gamma(t))\,\gamma''_i(t)+
\sum_{i,j}
\Big({\partial a_{ik}\over\partial x_j}-{1\over 2}
{\partial a_{ij}\over\partial x_k}\Big)(\gamma(t))\,\gamma'_i(t)\gamma'_j(t)=0,
\qquad 1\le k\le m.
$$

\section 5. Probl\`emes|

{\parindent1.4pc
\noindent{\bf 5.1.} On consid\`ere l'\'equation diff\'erentielle \`a variables
s\'epar\'ees
$$
(F_{\alpha})\qquad\frac{dy}{dt}=y^{\alpha}+1,\qquad \alpha>0.
$$

\item{(a)} Exprimer la solution g\'en\'erale de $(F_{\alpha})$ en introduisant la
fonction auxiliaire 
$$
G(y)=\int^y_0\frac{dx}{x^{\alpha}+1}
$$

\item{(b)} Plus pr\'ecis\'ement~:

\smallskip
\item{} $\bullet$ D\'eterminer (en distinguant les deux cas $0<\alpha\leq 1$ et
$\alpha>1$) le comportement $\phantom{\bullet}$~de $G(y)$ sur $[0,+\infty[$~;

\smallskip
\item{} $\bullet$ en d\'eduire dans chaque cas l'allure des solutions maximales de
$(F_{\alpha})$~;

\smallskip
\item{} $\bullet$ traiter compl\`etement et explicitement les deux cas $\alpha=1$ et
$\alpha=2$.


\bigskip
\noindent{\bf 5.2.} On consid\`ere l'\'equation diff\'erentielle 
$$
xy'-y^2+(2x+1)y=x^2+2x.
$$


\item{(a)} Poss\`ede-t-elle une solution particuli\`ere de type polyn\^ome~? En donner la
solution g\'en\'erale.

\medskip
\item{(b)} Quelle est l'\'equation de l'isocline de pente 0 dans le nouveau rep\`ere de
vecteurs de base $((1,1), (0,1))$~? Dessiner cette isocline en pr\'ecisant les tangentes
aux points d'abscisse 0 dans l'ancien rep\`ere.

\medskip
\item{(c)} Dessiner l'allure g\'en\'erale des solutions.

\medskip
\item{(d)} Soit $(x_0,y_0)$ un point de $\bR^2$. Combien passe-t-il de solutions maximales
de classe $C^1$ par $(x_0,y_0)$~? On pr\'ecisera l'intervalle de d\'efinition de ces
solutions et le cas \'ech\'eant on indiquera les solutions globales.



\bigskip
\noindent{\bf 5.3.} On consid\`ere l'\'equation diff\'erentielle
$$
\frac{dy}{dt}=y^2-(2x-1)y+x^2-x+1.
$$


\item{(a)} D\'eterminer explicitement les solutions de cette \'equation~; on
pourra commencer par chercher s'il existe des solutions polynomiales simples.

\medskip
\item{(b)} Montrer que les courbes int\'egrales maximales correspondant \`a des solutions
non polynomiales forment deux familles de courbes se d\'eduisant les unes des autres par
translations. Tracer celles de ces courbes qui sont asymptotes \`a l'axe $y'Oy$.


\bigskip
\noindent{\bf 5.4.} On consid\`ere l'\'equation diff\'erentielle (1) 
$y'^2=yy'+x$, o\`u $y$ est une fonction de $x$ \`a valeurs r\'eelles, 
de classe $C^1$ par morceaux.

\medskip

\item{(a)} Par quels points $(x,y)$ de $\bR^2$ passe-t-il une solution de (1)~? Faire un
graphique.

\medskip
\item{(b)} En param\'etrant (1) par $dy=tdx$, montrer que $y$ est solution
d'une \'equation diff\'erentielle (2) $f\big(y,t,\dfrac{dy}{dt}\big)=0$.

\medskip
\item{(c)} Int\'egrer (2) puis (1)~; on pourra poser $t=\tan\varphi$ avec 
$\varphi\in{}\big]-\dfrac{\pi}{2},\dfrac{\pi}{2}\big[$. On obtient une 
famille de courbes $C_{\lambda}$ d\'ependant d'un param\`etre $\lambda$.

\medskip
\item{(d)} Pour $\lambda=0$ pr\'eciser les limites quand
$\varphi\rightarrow\dfrac{\pi}{2}-0$ de $x,y,y/x$, puis pr\'eciser 
$\dfrac{dx}{d\varphi}$
et $\dfrac{dy}{d\varphi}$. Quelle est la norme euclidienne du vecteur
$\big(\dfrac{dx}{d\varphi},\dfrac{dy}{d\varphi}\big)$~? On se rappellera que
$\dfrac{dy}{dx}=\tan\varphi$.

\medskip
\item{(e)} On pose
$z(\varphi)=\sqrt{\big(\dfrac{dx}{d\varphi}\big)^2+
\big(\dfrac{dy}{d\varphi}\big)^2}$
pour $0\leq\varphi\leq\dfrac{\pi}{2}$. Etudier les variations de
$z(\varphi)$ puis tracer la courbe $C_0$.


\bigskip
\noindent{\bf 5.5.} On consid\`ere la famille de paraboles $(P_{\lambda})$ d'\'equation
$$
P_{\lambda}:x=y^2+\lambda y.
$$


\item{(a)} Montrer que ces courbes sont solutions d'une \'equation diff\'erentielle du
premier ordre que l'on pr\'ecisera.

\medskip
\item{(b)} D\'eterminer la famille des courbes orthogonales aux courbes $(P_{\lambda})$.


\bigskip
\noindent{\bf 5.6.} On consid\`ere la famille de courbes $(C_{\lambda})$ dans $\bR^2$
d\'efinies par l'\'equation
$$
x^2-y^2+\lambda y^3=0,
$$
o\`u $\lambda$ est un param\`etre r\'eel.

\medskip

\item{(a)} Tracer les courbes $(C_0)$, $(C_1)$ dans un rep\`ere orthonorm\'e $Oxy$
(unit\'e~: 4 cm). Quelle relation existe-t-il entre $(C_1)$ et $(C_{\lambda})$~?

\medskip
\item{(b)} Montrer que les courbes $(C_{\lambda})$ sont solutions d'une \'equation
diff\'erentielle du premier ordre.

\medskip
\item{(c)} D\'eterminer l'\'equation des trajectoires orthogonales aux courbes
$(C_{\lambda})$. Quelle est la nature de ces courbes~? Repr\'esenter la trajectoire
orthogonale passant par le point $(1,0)$ sur le m\^eme sch\'ema que $(C_0)$ et $(C_1)$.


\bigskip
\noindent{\bf 5.7.} On consid\`ere dans le plan euclidien $\bR^2$ la famille de courbes
$$
(C_{\lambda})\qquad x^4=y^4+\lambda x.
$$


\item{(a)} D\'eterminer l'\'equation diff\'erentielle v\'erifi\'ee par la
famille $(C_{\lambda})$.

\smallskip
\item{(b)} \'Ecrire l'\'equation diff\'erentielle des trajectoires
orthogonales aux courbes $(C_{\lambda})$.

\smallskip
\item{} En observant que cette \'equation est d'un type classique, d\'eterminer
l'\'equation des trajectoires orthogonales.

\bigskip
\noindent{\bf 5.8.} On consid\`ere le probl\`eme de Cauchy
$$
y'=t^2+y^2+1\,;\qquad y(0)=0.\leqno{\rm (P)}
$$
\item{(a)} Soient $T$ et $R$ deux r\'eels $>0$, et soit $\Omega(T,R)$ le 
rectangle d\'efini par les in\'egalit\'es
$$
0\leq t\leq T\,;\quad -R\leq y\leq R.
$$

\vbox{
\itemitem{$(\alpha)$} Montrer que si la condition
$$
T^2+R^2+1\leq R/T
$$
est v\'erifi\'ee, alors $\Omega(T,R)$ est un rectangle de 
s\'ecurit\'e pour~(P).}

\smallskip
\itemitem{$(\beta)$} Montrer que pour $T>0$ suffisamment petit, par exemple 
pour $T<T_0=\sqrt{\dfrac{\sqrt{2}-1}{2}}$, il existe $R>0$ tel que
$\Omega(T,R)$ soit un rectangle de s\'ecurit\'e pour~(P).

\smallskip
\itemitem{$(\gamma)$} Montrer que $\Omega(1/3,2)$ est un rectangle de 
s\'ecurit\'e pour (P), et en d\'eduire avec pr\'ecision que (P) admet
une solution $y$ et une seule sur l'intervalle $[0,1/3]$.

\medskip
\item{(b)} On va montrer que la solution $y$ de (P) mise en \'evidence
en (a)$\,(\gamma)$ ne se prolonge pas \`a $[0,+\infty[$. On introduit
\`a cet effet le probl\`eme de Cauchy auxiliaire
$$
z'=z^2+1,\qquad z(0)=0,\leqno{\rm (P_1)}
$$
\item{} o\`u $z$ d\'esigne une nouvelle fonction inconnue de $t$.

\smallskip
\itemitem{$(\alpha)$} D\'eterminer explicitement l'unique solution $z$ de
(P$_1$), et indiquer son intervalle de d\'efinition maximal.

\smallskip
\itemitem{$(\beta)$} Soit $[0,T]$ un intervalle sur lequel $y$ et $z$ soient 
simultan\'ement d\'efinies. Montrer que $u=y-z$ est solution d'un probl\`eme
de Cauchy
$$
u'=a(t)u+b(t)\,;\qquad u(0)=0\,;\leqno({\rm P}_2)
$$
o\`u $b$ v\'erifie $b(t)\geq 0$ pour tout $t$ dans $[0,T]$. En 
d\'eduire que $y(t)\geq z(t)$ pour tout $t$ dans $[0,T]$.

\smallskip
\itemitem{$(\gamma)$} D\'eduire de ce qui pr\'ec\`ede que si 
$T\geq\dfrac{\pi}{2}$, alors $y$ ne se prolonge certainement pas jusqu'\`a
$t=T$.

\smallskip
\itemitem{$(\delta)$} Tracer avec le maximum de pr\'ecision possible le graphe 
de $y$ sur son intervalle de d\'efinition maximal $[0,T_1[$ (forme du graphe
au voisinage de $0$~; sens de variation, convexit\'e~; asymptote~; 
etc$\ldots$).

\bigskip
\noindent{\bf 5.9${}^{**}$.} On appelle m\'etrique de Poincar\'e du disque unit\'e 
$D=\{|z|<1\}$ du plan complexe la m\'etrique riemannienne
$$
ds^2={|dz|^2\over (1-|z|^2)^2}={dx^2+dy^2\over(1-(x^2+y^2))^2},\qquad
z=x+i y.
$$
\item{(a)} Montrer (avec les notations du \S$\,$4.4, et en gardant les
fonctions complexes dans les calculs) que la diff\'erentielle de
l'\'energie est donn\'ee par
$$
E(\gamma)\cdot h=2\Re\int_a^b\Big({\gamma'(t)\over
\big(1-\gamma(t)\ol{\gamma(t)}\big)^2}\ol{h'(t)}
+2{\gamma(t)\gamma'(t)\ol{\gamma'(t)}\over
\big(1-\gamma(t)\ol{\gamma(t)}\big)^3}\ol{h(t)}\Big)dt.
$$
En d\'eduire que l'\'equation d'Euler-Lagrange des g\'eod\'esiques est
$$
\gamma''(t)+{2\gamma'(t)^2\ol{\gamma(t)}\over
1-\gamma(t)\ol{\gamma(t)}}=0.
$$

\item{(b)} Montrer que le chemin $\gamma(t)=\tanh(kt)$ (qui d\'ecrit le
diam\`etre $]-1,1[$ du disque) est solution de l'\'equation pour
tout $k\in\bR^*_+$.
\medskip

\item{(c)} Montrer que si $t\mapsto \gamma(t)$ est solution, alors 
$t\mapsto\lambda\gamma(t)$ est encore solution pour tout nombre complexe 
$\lambda$ de module $1$,
et \'egalement que $h_a\circ\gamma$ est solution, pour toute homographie 
complexe  $h_a$ de la forme
$$h_a(z)={z+a\over 1+\ol a z},\qquad\hbox{$a\in D$}.$$

\item{(d)} En utilisant un argument d'unicit\'e des solutions du probl\`eme
de Cauchy, montrer que les g\'eod\'esiques sont toutes donn\'ees par
$\gamma(t)=h_a(\lambda\tanh(kt))$, $a\in D$, $|\lambda|=1$, $k\in\bR_+^*$
[on pourra calculer $\gamma(0)$ et $\gamma'(0)$].

\item{(e)} Montrer que les trajectoires des g\'eod\'esiques sont les 
diam\`etres et les arcs de cercle orthogonaux au cercle unit\'e $|z|=1$.
\par}

\bye
% Local Variables:
% TeX-command-default: "TeX"
% End:
