Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/25

la différentielle de $z,$ prise en regardant $p$ et $q$ comme constants, on a

ϐ"$={\frac {\partial z}{\partial r}}.$

ϐ$'$ est le coefficient de $dq$ dans la différentielle de $y,$ prise en regardant $p$ et $z$ comme constants ; on aura donc ϐ$'$ en différenciant $y$ dans la supposition de $p$ constant, et en éliminant $dr$ au moyen de la différentielle de $z,$ prise en supposant $p$ constant et égalée à zéro ; on aura ainsi les deux équations

${\begin{aligned}dy&={\frac {\partial y}{\partial q}}dq+{\frac {\partial y}{\partial r}}dr,\\\\0&={\frac {\partial z}{\partial q}}dq+{\frac {\partial z}{\partial r}}dr,\end{aligned}}$

ce qui donne

$dy=dq{\frac {{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial r}}-{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial q}}}{\frac {\partial z}{\partial r}}}\,;$

partant

ϐ$'={\frac {{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial r}}-{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial q}}}{\frac {\partial z}{\partial r}}}.$

Enfin ϐ est le coefficient de $dp$ dans la différentielle de $x,$ prise en regardant $y$ et $z$ comme constants, ce qui donne les trois équations suivantes :

${\begin{aligned}dx&={\frac {\partial x}{\partial p}}dp+{\frac {\partial x}{\partial q}}dq+{\frac {\partial x}{\partial r}}dr,\\\\0&={\frac {\partial y}{\partial p}}dp+{\frac {\partial y}{\partial q}}dq+{\frac {\partial y}{\partial r}}dr,\\\\0&={\frac {\partial z}{\partial p}}dp+{\frac {\partial z}{\partial q}}dq+{\frac {\partial z}{\partial r}}dr,\\\\\end{aligned}}$

Si l’on fait

${\begin{aligned}\varepsilon &={\frac {\partial x}{\partial p}}{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial r}}-{\frac {\partial x}{\partial p}}{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial q}}\\\\&+{\frac {\partial x}{\partial q}}{\frac {\partial y}{\partial r}}{\frac {\partial z}{\partial p}}-{\frac {\partial x}{\partial q}}{\frac {\partial y}{\partial p}}{\frac {\partial z}{\partial r}}\\\\&+{\frac {\partial x}{\partial r}}{\frac {\partial y}{\partial p}}{\frac {\partial z}{\partial q}}-{\frac {\partial x}{\partial r}}{\frac {\partial y}{\partial q}}{\frac {\partial z}{\partial p}},\end{aligned}}$