74
MÉCANIQUE ANALYTIQUE.
dessus, et des valeurs semblables de
car, en faisant ces substitutions dans les termes
et ordonnant par rapport aux quantités
on aura
![{\displaystyle {\begin{aligned}&\left({\frac {\partial x}{\partial a}}{\frac {\partial a}{\partial x'}}+{\frac {\partial x}{\partial b}}{\frac {\partial b}{\partial x'}}+{\frac {\partial x}{\partial c}}{\frac {\partial c}{\partial x'}}+\ldots \right)\mathrm {X} dt\\+&\left({\frac {\partial x}{\partial a}}{\frac {\partial a}{\partial y'}}+{\frac {\partial x}{\partial b}}{\frac {\partial b}{\partial y'}}\,+{\frac {\partial x}{\partial c}}{\frac {\partial c}{\partial y'}}+\ldots \right)\mathrm {Y} dt\\+&\left({\frac {\partial x}{\partial a}}{\frac {\partial a}{\partial z'}}+{\frac {\partial x}{\partial b}}{\frac {\partial b}{\partial z'}}\,+{\frac {\partial x}{\partial c}}{\frac {\partial c}{\partial z'}}+\ldots \right)\mathrm {Z} dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d94ec7da52732d1746c258eeb2ecb3ecb9b01ca4)
Mais, en regardant d’abord
comme fonctions de
et ensuite
comme fonctions de
on a
![{\displaystyle {\begin{aligned}&dx={\frac {\partial x}{\partial a}}da+{\frac {\partial x}{\partial b}}db+{\frac {\partial x}{\partial c}}dc+{\frac {\partial x}{\partial h}}dh+\ldots ,\\&dy={\frac {\partial y}{\partial a}}da\,+{\frac {\partial y}{\partial b}}db\,+{\frac {\partial y}{\partial c}}dc\,+{\frac {\partial y}{\partial h}}dh+\ldots ,\\&.\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\&da={\frac {\partial a}{\partial x}}dx+{\frac {\partial a}{\partial y}}dy+{\frac {\partial a}{\partial z}}dz+{\frac {\partial a}{\partial x'}}dx'+\ldots ,\\&db={\frac {\partial b}{\partial x}}dx\,+{\frac {\partial b}{\partial y}}dy\,+{\frac {\partial b}{\partial z}}dz+{\frac {\partial b}{\partial x'}}dx'+\ldots ,\\&..\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9c87699166ceb2bb7f021679b9289bef6a8ed91)
Substituant dans l’expression de
ces valeurs de
on doit avoir des équations identiques ; par conséquent, il faudra que les termes affectés de
dans les expressions de
deviennent nuls ; ce qui donnera, par rapport à
les équations identiques
![{\displaystyle {\begin{aligned}&{\frac {\partial x}{\partial a}}{\frac {\partial a}{\partial x'}}+{\frac {\partial x}{\partial b}}{\frac {\partial b}{\partial x'}}+{\frac {\partial x}{\partial c}}{\frac {\partial c}{\partial x'}}+\ldots =0,\\&{\frac {\partial x}{\partial a}}{\frac {\partial a}{\partial y'}}\,+{\frac {\partial x}{\partial b}}{\frac {\partial b}{\partial y'}}\,+{\frac {\partial x}{\partial c}}{\frac {\partial c}{\partial y'}}+\ldots =0,\\&{\frac {\partial x}{\partial a}}{\frac {\partial a}{\partial z'}}\,+{\frac {\partial x}{\partial b}}{\frac {\partial b}{\partial z'}}\,+{\frac {\partial x}{\partial c}}{\frac {\partial c}{\partial z'}}+\ldots =0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b74ea43d815171f2d722603d97a79b366f07e54)
On aura donc simplement
![{\displaystyle dx={\frac {\partial x}{\partial t}}dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/750f0bc0b9ff8fe3605c3d0a52299ba4a1e4eeda)