302
MÉCANIQUE ANALYTIQUE.
les autres quantités
demeureront encore indéterminées.
On fera les mêmes substitutions dans l’équation (L) de l’article 15, laquelle équivaut aux trois équations (F) de l’article 10, et il est aisé de voir qu’elle se réduira à la forme suivante
![{\displaystyle {\begin{aligned}d\lambda -d\mathrm {V} =&\alpha dx+\beta dy+\gamma dz+z(\alpha 'dx+\beta 'dy+\gamma 'dz)\\&+z^{2}(\alpha ''dx+\beta ''dy+\gamma ''dz)+\ldots ,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fb1cf00aa7f148aae768aff4c2083bdb10e6ae8)
en faisant, pour abréger,
![{\displaystyle {\begin{aligned}\alpha =&{\frac {\partial p'}{\partial t}}+p'{\frac {\partial p'}{\partial x}}+q'{\frac {\partial p'}{\partial y}}+r'p'',\\\beta =&{\frac {\partial q'}{\partial t}}+p'{\frac {\partial q'}{\partial x}}+q'{\frac {\partial q'}{\partial y}}+r'q'',\\\gamma =&{\frac {\partial r'}{\partial t}}+p'{\frac {\partial r'}{\partial x}}+q'{\frac {\partial r'}{\partial y}}+r'r'',\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/790a8bb292a2ff3cf7a08785c4e688a37b26de47)
![{\displaystyle {\begin{aligned}\alpha '=&{\frac {\partial p''}{\partial t}}+p'{\frac {\partial p''}{\partial x}}+p''{\frac {\partial p'}{\partial x}}+q'{\frac {\partial p''}{\partial y}}+q''{\frac {\partial p'}{\partial y}}+2r'p'''+r''p'',\\\beta '=&{\frac {\partial q''}{\partial t}}+p'{\frac {\partial q''}{\partial x}}+p''{\frac {\partial q'}{\partial x}}+q'{\frac {\partial q''}{\partial y}}+q''{\frac {\partial q'}{\partial y}}+2r'q'''+r''q'',\\\gamma '=&{\frac {\partial r''}{\partial t}}+p'{\frac {\partial r''}{\partial x}}+p''{\frac {\partial r'}{\partial x}}+q'{\frac {\partial r''}{\partial y}}+q''{\frac {\partial r'}{\partial y}}+2r'r'''+r''r'',\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3a25099881b294e314500bd8b8e94900ef1f95e)
et ainsi de suite.
Donc, pour que le second membre de cette équation soit intégrable, il faudra que les quantités
![{\displaystyle \alpha dx+\beta dy,\quad \gamma dz+z(\alpha 'dx+\beta 'dy),\quad \gamma 'zdz+z^{2}(\alpha ''dx+\beta ''dy),\quad \ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9490c062b59b52d2e04d69973613f03a1986fced)
soient chacune intégrables en particulier.
Si donc on dénote par
une fonction de
sans
on aura ces conditions
![{\displaystyle \alpha ={\frac {\partial \omega }{\partial x}},\quad \beta ={\frac {\partial \omega }{\partial y}},\quad \alpha '={\frac {\partial \gamma }{\partial x}},\quad \beta '={\frac {\partial \gamma }{\partial y}},\quad \alpha ''={\frac {1}{2}}{\frac {\partial \gamma '}{\partial x}},\quad \beta ''={\frac {1}{2}}{\frac {\partial \gamma '}{\partial y}},\quad \ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4a4cc88c3d26af626cd45f4ae85aab5f705ae46)
Alors l’équation intégrée donnera
![{\displaystyle \lambda =\mathrm {V} +\omega +\gamma z+{\frac {1}{2}}\gamma 'z^{2}+\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49df96a7e3a2c41439cd5c475e4e3adb298f6408)