76
MÉCANIQUE ANALYTIQUE.
et, ces valeurs étant substituées dans l’expression de
de l’article 58, à la place de
elle deviendra
![{\displaystyle {\begin{aligned}da=&+\left({\frac {\partial a}{\partial x'}}{\frac {\partial a}{\partial x}}+{\frac {\partial a}{\partial y'}}{\frac {\partial a}{\partial y}}+{\frac {\partial a}{\partial z'}}{\frac {\partial a}{\partial z}}\right){\frac {\partial \Omega }{\partial a}}dt\\&+\left({\frac {\partial a}{\partial x'}}{\frac {\partial b}{\partial x}}+{\frac {\partial a}{\partial y'}}{\frac {\partial b}{\partial y}}+{\frac {\partial a}{\partial z'}}{\frac {\partial b}{\partial z}}\right){\frac {\partial \Omega }{\partial b}}dt\\&+\left({\frac {\partial a}{\partial x'}}{\frac {\partial c}{\partial x}}+{\frac {\partial a}{\partial y'}}{\frac {\partial c}{\partial y}}+{\frac {\partial a}{\partial z'}}{\frac {\partial c}{\partial z}}\right){\frac {\partial \Omega }{\partial c}}dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b3c560ec3b61a88cdd20b41ffec34bc1295b9cb)
On peut faire disparaître de cette expression les termes multipliés par
par la considération que,
ne contenantpointles variables
on a
![{\displaystyle {\begin{aligned}{\frac {\partial \Omega }{\partial x'}}=&{\frac {\partial \Omega }{\partial a}}{\frac {\partial a}{\partial x'}}+{\frac {\partial \Omega }{\partial b}}{\frac {\partial b}{\partial x'}}+{\frac {\partial \Omega }{\partial c}}{\frac {\partial c}{\partial x'}}+\ldots =0,\\{\frac {\partial \Omega }{\partial y'}}=&{\frac {\partial \Omega }{\partial a}}{\frac {\partial a}{\partial y'}}+{\frac {\partial \Omega }{\partial b}}{\frac {\partial b}{\partial y'}}\,+{\frac {\partial \Omega }{\partial c}}{\frac {\partial c}{\partial y'}}+\ldots =0,\\{\frac {\partial \Omega }{\partial z'}}=&{\frac {\partial \Omega }{\partial a}}{\frac {\partial a}{\partial z'}}\,+{\frac {\partial \Omega }{\partial b}}{\frac {\partial b}{\partial z'}}\,+{\frac {\partial \Omega }{\partial c}}{\frac {\partial c}{\partial z'}}+\ldots =0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc185db1434d7b37890e01f553a09645b1ac256)
Donc, si l’on soustrait de la valeur de
la quantité
![{\displaystyle \left({\frac {\partial \Omega }{\partial x'}}{\frac {\partial a}{\partial x}}+{\frac {\partial \Omega }{\partial y'}}{\frac {\partial a}{\partial y}}+{\frac {\partial \Omega }{\partial z'}}{\frac {\partial a}{\partial z}}\right)dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a7d5cfa9b6e6a5afcc3f5e01c8a6f0efc781344)
![{\displaystyle {\begin{aligned}da=&+\left({\frac {\partial a}{\partial x'}}{\frac {\partial b}{\partial x}}+{\frac {\partial a}{\partial y'}}{\frac {\partial b}{\partial y}}+{\frac {\partial a}{\partial z'}}{\frac {\partial b}{\partial z}}-{\frac {\partial a}{\partial x}}{\frac {\partial b}{\partial x'}}-{\frac {\partial a}{\partial y}}{\frac {\partial b}{\partial y'}}-{\frac {\partial a}{\partial z}}{\frac {\partial b}{\partial z'}}\right){\frac {\partial \Omega }{\partial b}}dt\\&+\left({\frac {\partial a}{\partial x'}}{\frac {\partial c}{\partial x}}+{\frac {\partial a}{\partial y'}}{\frac {\partial c}{\partial y}}+{\frac {\partial a}{\partial z'}}{\frac {\partial c}{\partial z}}-{\frac {\partial a}{\partial x}}{\frac {\partial c}{\partial x'}}-{\frac {\partial a}{\partial y}}{\frac {\partial c}{\partial y'}}-{\frac {\partial a}{\partial z}}{\frac {\partial c}{\partial z'}}\right){\frac {\partial \Omega }{\partial c}}dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1117ff3cedad2dea79c6cf275dada07d21ee834c)
Cette expression de
est, en apparence, plus compliquée que la formule primitive d’où nous sommes partis ; mais elle a, d’un autre côté, le grand avantage que les coefficients des différences partielles
deviennent indépendants du temps
après la substitution des valeurs de
en
et
données par le mou-