ou
![{\displaystyle dy=-{\frac {{\cfrac {\partial \theta }{\partial x}}dx}{\cfrac {\partial \theta }{\partial y}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e60626d39be4f464a6f2af4936e8f2f8a95c296f)
on a ensuite
![{\displaystyle 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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1bd0257aa101802acdbed23e8f9f992d89716cb)
donc
![{\displaystyle {\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}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59887a98685cc4af15aa625ebe57890a818e9103)
et changeant
en
et réciproquement, on aura
![{\displaystyle {\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}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c942823592f33fd3f7708590fd0fac50dcfb7de)
éliminant ensuite
et
au moyen des équations (2) et (3), on aura
en fonction de
![{\displaystyle 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}}}\ }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aacb1d8ea4d51a2deed927de0eb31e0e542064f0)
et
![{\displaystyle \ {\frac {\partial ^{3}\theta }{\partial y^{3}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35d75f8b5b2090667d20dbfed761623edd205365)
on aura par un procédé semblable les valeurs de
![{\displaystyle {\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 \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1e8eb0a95b312f4c2a20c4d96b9418156cb860b)
l’équation
deviendra ainsi fonction de
![{\displaystyle x,\ \ y,\ \ {\frac {\partial \varpi }{\partial y}},\ \ {\frac {\partial ^{2}\varpi }{\partial y^{2}}},\ \ ,\ldots ,\ \ {\frac {\partial \theta }{\partial y}},\ \ \ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a67a343616572bebe6718375ca588f91aa1fa3d)
et en réduisant tous les termes au même dénominateur, et faisant égale à zéro la somme de tous les numérateurs, on aura entre ![{\displaystyle x,\ \ y,\ \ {\frac {\partial \varpi }{\partial y}},\ \ {\frac {\partial ^{2}\varpi }{\partial y^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e06e425c37633c1b622f69530279a1af691757)
une équation rationnelle et entière par rapport aux