Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/129

121

SECONDE PARTIE. — SECTION VII.

tiendra d’abord les termes

$\mathrm {(m+m')} \int \mathrm {R} 'd\rho '+\mathrm {m} ''\int \mathrm {R} ''_{_{'}}d\rho ''_{_{'}}+\mathrm {m} '''\int \mathrm {R} '''_{_{'}}d\rho '''_{_{'}}+\ldots \,;$

ensuite elle contiendra les termes

${\begin{aligned}&\mathrm {\left(m''R''{\frac {\partial \rho ''}{\partial \xi ''}}+m'''R'''{\frac {\partial \rho '''}{\partial \xi '''}}+\ldots \right)} \xi '\\+&\mathrm {\left(m''R''{\frac {\partial \rho ''}{\partial \eta ''}}+m'''R'''{\frac {\partial \rho '''}{\partial \eta '''}}+\ldots \right)} \eta '\\+&\mathrm {\left(m''R''{\frac {\partial \rho ''}{\partial \zeta ''}}+m'''R'''{\frac {\partial \rho '''}{\partial \zeta '''}}+\ldots \right)} \zeta '\\\end{aligned}}$

Or on a

${\frac {\partial \rho ''}{\partial \xi ''}}={\frac {\xi ''}{\rho ''}},\qquad {\frac {\partial \rho ''}{\partial \eta ''}}={\frac {\eta ''}{\rho ''}},\qquad {\frac {\partial \rho ''}{\partial \zeta ''}}={\frac {\zeta ''}{\rho ''}}\,;$

et comme

$\rho _{_{'}}^{''2}=(\xi ''-\xi ')^{2}+(\eta ''-\eta ')^{2}+(\zeta ''-\zeta ')^{2},$

On aura

${\frac {\partial \rho ''}{\partial \xi ''}}\xi '+{\frac {\partial \rho ''}{\partial \eta ''}}\eta '+{\frac {\partial \rho ''}{\partial \zeta ''}}\zeta '={\frac {\rho '^{2}+\rho ''^{2}-\rho _{_{'}}^{''2}}{2\rho ''}},$

et de même

${\frac {\partial \rho '''}{\partial \xi '''}}\xi '+{\frac {\partial \rho '''}{\partial \eta '''}}\eta '+{\frac {\partial \rho '''}{\partial \zeta '''}}\zeta '={\frac {\rho '^{2}+\rho '''^{2}-\rho _{_{'}}^{'''2}}{2\rho '''}},$

et ainsi des autres expressions semblables. Donc, nommant $\mathrm {V} '$ l’intégrale totale, on aura

${\begin{aligned}\mathrm {V} '=&\mathrm {(m+m')} \int \mathrm {R} 'd\rho '+\mathrm {m} ''\int \mathrm {R} ''_{_{'}}d\rho ''_{_{'}}+\mathrm {m} '''\int \mathrm {R} '''_{_{'}}d\rho '''_{_{'}}+\ldots \\&+\mathrm {m''R''} {\frac {\rho '^{2}+\rho ''^{2}-\rho _{_{'}}^{''2}}{2\rho ''}}+\mathrm {m'''R'''} {\frac {\rho '^{2}+\rho '''^{2}-\rho _{_{'}}^{'''2}}{2\rho '''}}+\ldots .\end{aligned}}$

Ainsi, après la transformation des coordonnées, les termes multipliés par $\delta \xi$ se réduiront à ${\frac {\delta \mathrm {V} '}{\delta \xi }}\delta \xi \,;$ et, comme on suppose les nouvelles coordonnées $\xi ,\,\psi ,\,\varphi$ indépendantes, chacune d’elles, comme $\xi ,$ donnera une équation de la forme

$d{\frac {\delta \mathrm {T} '}{\delta d\xi }}-{\frac {\delta \mathrm {T} '}{\delta \xi }}+{\frac {\delta \mathrm {V} '}{\delta \xi }}=0.$