121
SECONDE PARTIE. — SECTION VII.
tiendra d’abord les termes
![{\displaystyle \mathrm {(m+m')} \int \mathrm {R} 'd\rho '+\mathrm {m} ''\int \mathrm {R} ''_{_{'}}d\rho ''_{_{'}}+\mathrm {m} '''\int \mathrm {R} '''_{_{'}}d\rho '''_{_{'}}+\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8771acdbfc92f8babc9dac2026a6c89e277b8f1c)
ensuite elle contiendra les termes
![{\displaystyle {\begin{aligned}&\mathrm {\left(m''R''{\frac {\partial \rho ''}{\partial \xi ''}}+m'''R'''{\frac {\partial \rho '''}{\partial \xi '''}}+\ldots \right)} \xi '\\+&\mathrm {\left(m''R''{\frac {\partial \rho ''}{\partial \eta ''}}+m'''R'''{\frac {\partial \rho '''}{\partial \eta '''}}+\ldots \right)} \eta '\\+&\mathrm {\left(m''R''{\frac {\partial \rho ''}{\partial \zeta ''}}+m'''R'''{\frac {\partial \rho '''}{\partial \zeta '''}}+\ldots \right)} \zeta '\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d5b3ca6f35a76e71056092b8e20df2dc3faf12d)
Or on a
![{\displaystyle {\frac {\partial \rho ''}{\partial \xi ''}}={\frac {\xi ''}{\rho ''}},\qquad {\frac {\partial \rho ''}{\partial \eta ''}}={\frac {\eta ''}{\rho ''}},\qquad {\frac {\partial \rho ''}{\partial \zeta ''}}={\frac {\zeta ''}{\rho ''}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9e3e14e55c35e5ce6fb131769a41f8b7424e725)
et comme
![{\displaystyle \rho _{_{'}}^{''2}=(\xi ''-\xi ')^{2}+(\eta ''-\eta ')^{2}+(\zeta ''-\zeta ')^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89ee6819dd0877c26364d5e3a577c77a4f9c0327)
On aura
![{\displaystyle {\frac {\partial \rho ''}{\partial \xi ''}}\xi '+{\frac {\partial \rho ''}{\partial \eta ''}}\eta '+{\frac {\partial \rho ''}{\partial \zeta ''}}\zeta '={\frac {\rho '^{2}+\rho ''^{2}-\rho _{_{'}}^{''2}}{2\rho ''}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03e5402a4b9a288115873c2c80c1a705bfd62fcd)
et de même
![{\displaystyle {\frac {\partial \rho '''}{\partial \xi '''}}\xi '+{\frac {\partial \rho '''}{\partial \eta '''}}\eta '+{\frac {\partial \rho '''}{\partial \zeta '''}}\zeta '={\frac {\rho '^{2}+\rho '''^{2}-\rho _{_{'}}^{'''2}}{2\rho '''}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7beee0fc524d84842376da45c9c21f03a06591e)
et ainsi des autres expressions semblables. Donc, nommant
l’intégrale totale, on aura
![{\displaystyle {\begin{aligned}\mathrm {V} '=&\mathrm {(m+m')} \int \mathrm {R} 'd\rho '+\mathrm {m} ''\int \mathrm {R} ''_{_{'}}d\rho ''_{_{'}}+\mathrm {m} '''\int \mathrm {R} '''_{_{'}}d\rho '''_{_{'}}+\ldots \\&+\mathrm {m''R''} {\frac {\rho '^{2}+\rho ''^{2}-\rho _{_{'}}^{''2}}{2\rho ''}}+\mathrm {m'''R'''} {\frac {\rho '^{2}+\rho '''^{2}-\rho _{_{'}}^{'''2}}{2\rho '''}}+\ldots .\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1632bf974b129df98a9e66a35474e215dbcd26c1)
Ainsi, après la transformation des coordonnées, les termes multipliés par
se réduiront à
et, comme on suppose les nouvelles coordonnées
indépendantes, chacune d’elles, comme
donnera une équation de la forme
![{\displaystyle d{\frac {\delta \mathrm {T} '}{\delta d\xi }}-{\frac {\delta \mathrm {T} '}{\delta \xi }}+{\frac {\delta \mathrm {V} '}{\delta \xi }}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81d226897fe6d2dd295edb0eefada16c2d668d7f)