Page:Joseph Louis de Lagrange - Œuvres, Tome 11.djvu/367

349

SECONDE PARTIE. — SECTION V.

On aura pareillement

$\delta \xi \,d\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}=d\left(\delta \xi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}\right)-\delta \xi '\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}dt,$

et ainsi des autres formules semblables.

Par le moyen de ces transformations, l’équation précédente deviendra de cette forme

${\begin{aligned}d&\left\{{\begin{aligned}&\Delta \xi \,\delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}+\Delta \psi \,\delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}+\Delta \varphi \,\delta {\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots \\-&\delta \xi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}-\delta \psi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}-\delta \varphi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \varphi '}}-\ldots \end{aligned}}\right\}\\\\-&\left\{{\begin{aligned}&\Delta \xi \,\delta {\frac {\partial \mathrm {Z} }{\partial \xi }}+\Delta \psi \,\delta {\frac {\partial \mathrm {Z} }{\partial \psi }}+\Delta \varphi \,\delta {\frac {\partial \mathrm {Z} }{\partial \varphi }}+\ldots \\+&\Delta \xi '\delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}+\Delta \psi '\delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}+\Delta \varphi '\delta {\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots \end{aligned}}\right\}dt\\\\+&\left\{{\begin{aligned}&\delta \xi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \xi }}+\delta \psi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \psi }}+\delta \varphi \,\Delta {\frac {\partial \mathrm {Z} }{\partial \varphi }}+\ldots \\+&\delta \xi '\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}+\delta \psi '\Delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}+\delta \varphi '\Delta {\frac {\partial \mathrm {Z} }{\partial \varphi '}}+\ldots \end{aligned}}\right\}dt=0.\end{aligned}}$

6. Or, si l’on développe les expressions $\delta {\frac {\partial \mathrm {Z} }{\partial \xi }},\,\delta {\frac {\partial \mathrm {Z} }{\partial \xi '}},\ldots ,$ ainsi que les expressions semblables $\Delta {\frac {\partial \mathrm {Z} }{\partial \xi }},\,\Delta {\frac {\partial \mathrm {Z} }{\partial \xi '}},\ldots ,$ en regardant $\mathrm {Z}$ comme fonction de $\xi ,\,\psi ,\,\varphi ,\ldots$ et de $\xi ',\,\psi ',\,\varphi ',\ldots ,$ il est facile de voir que les termes multipliés par $dt$ dans l’équation précédente se détruisent mutuellement. En effet, on a

${\begin{alignedat}{5}\delta {\frac {\partial \mathrm {Z} }{\partial \xi }}&=&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi ^{2}}}\delta \xi &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi \,\partial \psi }}\delta \psi +\ldots &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi \,\partial \xi '}}\delta \xi '&+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi \,\partial \psi '}}\delta \psi '+\ldots ,\\\delta {\frac {\partial \mathrm {Z} }{\partial \psi }}&=&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi \,\partial \psi }}\delta \xi &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \psi ^{2}}}\delta \psi +\ldots &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \psi \,\partial \xi '}}\delta \xi '&+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \psi \,\partial \psi '}}\delta \psi '+\ldots ,\\\end{alignedat}}$

,

${\begin{alignedat}{6}&\delta {\frac {\partial \mathrm {Z} }{\partial \xi '}}&=&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi \partial \xi '}}\delta \xi &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \psi \,\partial \xi '}}\delta \psi +\ldots &+&\ \ {\frac {\partial ^{2}\mathrm {Z} }{\partial \xi '^{2}}}\ \ \delta \xi '&+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi '\partial \psi '}}\delta \psi '&+&\ldots ,\\&\delta {\frac {\partial \mathrm {Z} }{\partial \psi '}}&=&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi \,\partial \psi '}}\delta \xi &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \psi \partial \psi '}}\delta \psi +\ldots &+&{\frac {\partial ^{2}\mathrm {Z} }{\partial \xi '\partial \psi '}}\delta \xi '&+&\ \ {\frac {\partial ^{2}\mathrm {Z} }{\partial \psi '^{2}}}\ \ \delta \psi '&+&\ldots ,\\\end{alignedat}}$