Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/52

ou

$dy=-{\frac {{\cfrac {\partial \theta }{\partial x}}dx}{\cfrac {\partial \theta }{\partial y}}}\,;$

on a ensuite

$d\varpi ={\frac {\partial \varpi }{\partial x}}dx+{\frac {\partial \varpi }{\partial y}}dy={\frac {\partial \varpi }{\partial x}}dx-{\frac {{\cfrac {\partial \varpi }{\partial y}}{\cfrac {\partial \theta }{\partial x}}dx}{\cfrac {\partial \theta }{\partial y}}},$

donc

${\frac {\partial x}{\partial \varpi }}={\frac {\cfrac {\partial \theta }{\partial y}}{{\cfrac {\partial \varpi }{\partial x}}{\cfrac {\partial \theta }{\partial y}}-{\cfrac {\partial \varpi }{\partial y}}{\cfrac {\partial \theta }{\partial x}}}}\,;$

et changeant $x$ en $y,$ et réciproquement, on aura

${\frac {\partial y}{\partial \varpi }}={\frac {\cfrac {\partial \theta }{\partial x}}{{\cfrac {\partial \varpi }{\partial y}}{\cfrac {\partial \theta }{\partial x}}-{\cfrac {\partial \varpi }{\partial x}}{\cfrac {\partial \theta }{\partial y}}}}\,;$

éliminant ensuite ${\frac {\partial \varpi }{\partial x}}$ et ${\frac {\partial \theta }{\partial x}}$ au moyen des équations (2) et (3), on aura ${\frac {\partial m}{\partial \varpi }}$ en fonction de

$x,\ \ y,\ \ {\frac {\partial \varpi }{\partial y}},\ \ {\frac {\partial ^{2}\varpi }{\partial y^{2}}},\ \ {\frac {\partial ^{3}\varpi }{\partial y^{3}}},\ \ {\frac {\partial \theta }{\partial y}},\ \ {\frac {\partial ^{2}\theta }{\partial y^{2}}}\$ et $\ {\frac {\partial ^{3}\theta }{\partial y^{3}}}\,;$

on aura par un procédé semblable les valeurs de

${\frac {\partial m}{\partial \theta }},\ \ {\frac {\partial ^{2}m}{\partial \varpi ^{2}}},\ \ {\frac {\partial ^{2}m}{\partial \varpi \partial \theta }},\ \ {\frac {\partial ^{2}m}{\partial \theta ^{2}}},\ \ \ldots ,\ \ {\frac {\partial m}{\partial \varpi }},\ \ \ldots ,\ \ {\frac {\partial l}{\partial \varpi }},\ \ \ldots \,;$

l’équation $\mathrm {K} =0$ deviendra ainsi fonction de

$x,\ \ y,\ \ {\frac {\partial \varpi }{\partial y}},\ \ {\frac {\partial ^{2}\varpi }{\partial y^{2}}},\ \ ,\ldots ,\ \ {\frac {\partial \theta }{\partial y}},\ \ \ldots ,$

et en réduisant tous les termes au même dénominateur, et faisant égale à zéro la somme de tous les numérateurs, on aura entre $x,\ \ y,\ \ {\frac {\partial \varpi }{\partial y}},\ \ {\frac {\partial ^{2}\varpi }{\partial y^{2}}},$$\ldots ,\ \ {\frac {\partial \theta }{\partial y}},\ \ \ldots$ une équation rationnelle et entière par rapport aux