78
MÉCANIQUE ANALYTIQUE.
On trouve de la même manière
![{\displaystyle {\begin{aligned}d{\frac {\partial a}{\partial y}}=&\left({\frac {\partial a}{\partial x'}}{\frac {\partial ^{2}\mathrm {V} }{\partial x\partial y}}+{\frac {\partial a}{\partial y'}}{\frac {\partial ^{2}\mathrm {V} }{\partial y^{2}}}+{\frac {\partial a}{\partial z'}}{\frac {\partial ^{2}\mathrm {V} }{\partial y\partial z}}\right)dt,\\d{\frac {\partial a}{\partial z}}=&\left({\frac {\partial a}{\partial x'}}{\frac {\partial ^{2}\mathrm {V} }{\partial x\partial z}}+{\frac {\partial a}{\partial y'}}{\frac {\partial ^{2}\mathrm {V} }{\partial x\partial z}}+{\frac {\partial a}{\partial z'}}{\frac {\partial ^{2}\mathrm {V} }{\partial z^{2}}}\right)dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/218f10f90cf385e99641f4b462c0e74bc893141c)
On aura ensuite
![{\displaystyle d{\frac {\partial a}{\partial x'}}=\left\{{\begin{aligned}&{\frac {\partial ^{2}a}{\partial t\partial x'}}+{\frac {\partial ^{2}a}{\partial x\partial x'}}x'+{\frac {\partial ^{2}a}{\partial y\partial x'}}y'+{\frac {\partial ^{2}a}{\partial z\partial x'}}z'\\&\quad -{\frac {\partial ^{2}a}{\partial x'^{2}}}{\frac {\partial \mathrm {V} }{\partial x}}-{\frac {\partial ^{2}a}{\partial x'\partial y'}}{\frac {\partial \mathrm {V} }{\partial y}}-{\frac {\partial ^{2}a}{\partial x'\partial z'}}{\frac {\partial \mathrm {V} }{\partial z}}\end{aligned}}\right\}dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1efa8cb571d36ae4d8a3846ddb2f8b03975a510d)
Mais, en faisant varier
dans l’équation identique
et observant que la fonction
est supposée ne pas contenir les variables
on a
![{\displaystyle {\begin{aligned}&{\frac {\partial ^{2}a}{\partial t\partial x'}}+{\frac {\partial ^{2}a}{\partial x\partial x'}}x'+{\frac {\partial ^{2}a}{\partial y\partial x'}}y'+{\frac {\partial ^{2}a}{\partial z\partial x'}}z'\\&\quad -{\frac {\partial ^{2}a}{\partial x'^{2}}}{\frac {\partial \mathrm {V} }{\partial x}}-{\frac {\partial ^{2}a}{\partial x'\partial y'}}{\frac {\partial \mathrm {V} }{\partial y}}-{\frac {\partial ^{2}a}{\partial x'\partial z'}}{\frac {\partial \mathrm {V} }{\partial z}}=0\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56d5c73f56f66c9bec999e8946f481844bdae1fb)
donc on aura simplement
![{\displaystyle d{\frac {\partial a}{\partial x'}}=-{\frac {\partial a}{\partial x}}dt,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c84a4b4ca8b04ca9a30aa298aee2206562f26d2)
et l’on trouvera de la même manière
![{\displaystyle {\begin{aligned}d{\frac {\partial a}{\partial y'}}=&-{\frac {\partial a}{\partial y}}dt\\d{\frac {\partial a}{\partial z'}}=&-{\frac {\partial a}{\partial z}}dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10be01b9cf59775baee3ce91754b9c7e17e32032)
On aura des expressions semblables pour les différentielles
en changeant seulement
en
et ainsi pour les autres quantités semblables.
Si maintenant on différentie le coefficient de
dans l’expression de
de l’article 60, et qu’on y substitue les valeurs qu’on vient de