la différentielle de
prise en regardant
et
comme constants, on a
ϐ"
![{\displaystyle ={\frac {\partial z}{\partial r}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb65cd72d322daabe9d612e48339ef490ed3b3dd)
ϐ
est le coefficient de
dans la différentielle de
prise en regardant
et
comme constants ; on aura donc ϐ
en différenciant
dans la supposition de
constant, et en éliminant
au moyen de la différentielle de
prise en supposant
constant et égalée à zéro ; on aura ainsi les deux équations
![{\displaystyle {\begin{aligned}dy&={\frac {\partial y}{\partial q}}dq+{\frac {\partial y}{\partial r}}dr,\\\\0&={\frac {\partial z}{\partial q}}dq+{\frac {\partial z}{\partial r}}dr,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a83aae27772255019092b756dc4d8b25aefd9c9e)
ce qui donne
![{\displaystyle dy=dq{\frac {{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial r}}-{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial q}}}{\frac {\partial z}{\partial r}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6293d0714cbb50359fec8f71b26a587d33070861)
partant
ϐ
![{\displaystyle '={\frac {{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial r}}-{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial q}}}{\frac {\partial z}{\partial r}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819f46043fb02968280312f36eaa927bc049147d)
Enfin ϐ est le coefficient de
dans la différentielle de
prise en regardant
et
comme constants, ce qui donne les trois équations suivantes :
![{\displaystyle {\begin{aligned}dx&={\frac {\partial x}{\partial p}}dp+{\frac {\partial x}{\partial q}}dq+{\frac {\partial x}{\partial r}}dr,\\\\0&={\frac {\partial y}{\partial p}}dp+{\frac {\partial y}{\partial q}}dq+{\frac {\partial y}{\partial r}}dr,\\\\0&={\frac {\partial z}{\partial p}}dp+{\frac {\partial z}{\partial q}}dq+{\frac {\partial z}{\partial r}}dr,\\\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53e58bb544fc40efd457ea32a9e0052cadd8d740)
Si l’on fait
![{\displaystyle {\begin{aligned}\varepsilon &={\frac {\partial x}{\partial p}}{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial r}}-{\frac {\partial x}{\partial p}}{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial q}}\\\\&+{\frac {\partial x}{\partial q}}{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial p}}-{\frac {\partial x}{\partial q}}{\frac {\partial y}{\partial p}}{\frac {\partial z}{\partial r}}\\\\&+{\frac {\partial x}{\partial r}}{\frac {\partial y}{\partial p}}{\frac {\partial z}{\partial q}}-{\frac {\partial x}{\partial r}}{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial p}},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4bbe5a384944f5dd1db663a020b3d65d93f51d1)