36
INTÉGRALES
![{\displaystyle {\frac {\left(1+y'^{2}\right)x''-x'y'y''}{(1+x'^{2}+y'^{2})^{\frac {3}{2}}}}=0,\qquad {\frac {\left(1+x'^{2}\right)y''-x'y'x''}{(1+x'^{2}+y'^{2})^{\frac {3}{2}}}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/45cb3f655d651b39e39f6bd5cf07d5f1e522196e)
qu’on pourra mettre ensuite sous cette forme
![{\displaystyle {\frac {\left(1+x'^{2}+y'^{2}\right)x''-(x'x''+y'y'')x'}{(1+x'^{2}+y'^{2})^{\frac {3}{2}}}}=0,\ {\frac {\left(1+x'^{2}+y'^{2}\right)y''-(x'x''+y'y'')y'}{(1+x'^{2}+y'^{2})^{\frac {3}{2}}}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e60429df4c95da0cccef88cac1c1ee08cecbc464)
ou, en continuant d’employer les notations de Lagrange,
![{\displaystyle {\frac {\left(1+x'^{2}+y'^{2}\right)x''-\left(1+x'^{2}+y'^{2}\right)'x'}{(1+x'^{2}+y'^{2})^{\frac {3}{2}}}}=0,\ {\frac {\left(1+x'^{2}+y'^{2}\right)y''-\left(1+x'^{2}+y'^{2}\right)'y'}{(1+x'^{2}+y'^{2})^{\frac {3}{2}}}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a626423ef165f423ad774897c78de0942d59c5f)
ou encore
![{\displaystyle {\frac {x''{\sqrt {1+x'^{2}+y'^{2}}}-x'\left({\sqrt {1+x'^{2}+y'^{2}}}\right)'}{1+x'^{2}+y'^{2}}}=0,\ {\frac {y''{\sqrt {1+x'^{2}+y'^{2}}}-y'\left({\sqrt {1+x'^{2}+y'^{2}}}\right)'}{1+x'^{2}+y'^{2}}}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76bcbdb099d4d5d8fa9c3a64e173e91a7bfe205d)
ou enfin
![{\displaystyle \left({\frac {x'}{\sqrt {1+x'^{2}+y'^{2}}}}\right)'=0,\qquad \left({\frac {y'}{\sqrt {1+x'^{2}+y'^{2}}}}\right)'=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a65be58a4c3275938981d29ed31d03957822a98)
ce qui donne
![{\displaystyle {\frac {x'}{\sqrt {1+x'^{2}+y'^{2}}}}=A,\qquad {\frac {y'}{\sqrt {1+x'^{2}+y'^{2}}}}=B.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e7dd9a0d6c59cb9e60fcada49974b6cc353d7b8)
En considérant, dans ces équations,
et
comme deux inconnues, on en tire, en transformant les constantes,
![{\displaystyle x'={\frac {A}{\sqrt {1-A^{2}-B^{2}}}}=M,\qquad y'={\frac {B}{\sqrt {1-A^{2}-B^{2}}}}=N,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea6973d2f281212d5cba7c371736f94820ba7d3)
d’où enfin
![{\displaystyle x=Mz+G,\qquad y=Nz+H,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e127f63e2ebf2fa2a845ee5aa969254049e3ae)