Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/248

donc on aura les deux équations suivantes

${\begin{aligned}\delta ^{2}u^{2}dt^{2}=&(rdr)^{2}\left(r'^{2}u'^{2}dt^{2}-r'^{2}dr'^{2}\right)+d\mathrm {V} ^{2}\left(r^{2}u'^{2}dt^{2}-d\mathrm {V} '^{2}\right)\\&+\left(v''dt^{2}\right)^{2}\left(r^{2}r'^{2}-p''^{2}\right)+2rdrd\mathrm {V} \left(r'dr'd\mathrm {V} '-p''u'^{2}dt^{2}\right)\\&+2rdrv''dt^{2}\left(p''r'dr'-r'^{2}d\mathrm {V} '\right)+2d\mathrm {V} v''dt^{2}\left(p''d\mathrm {V} '-r^{2}r'dr'\right),\\\\\delta ^{2}r^{2}=&r^{4}\left(r'^{2}u'^{2}dt^{2}-r'^{2}dr'^{2}\right)+p''^{2}\left(r^{2}u'^{2}dt^{2}-d\mathrm {V} '^{2}\right)\\&+d\mathrm {V} '^{2}\left(r^{2}r'^{2}-p''^{2}\right)+2r^{2}p''\left(r'dr'd\mathrm {V} '-p''u'^{2}dt^{2}\right)\\&+2r^{2}d\mathrm {V} '\left(p''r'dr'-r'^{2}d\mathrm {V} '\right)+2p''d\mathrm {V} '\left(p''d\mathrm {V} '-r^{2}r'dr'\right).\end{aligned}}$

D’où, chassant $\delta _{2},$ on aura une équation entre les seules quantités connues $r^{2},r'^{2},\ldots .$

X.

Si l’on tire de la dernière équation la valeur de $\delta ^{2},$ on aura, en réduisant et effaçant ce qui se détruit,

$\delta ^{2}=r'^{2}\left(r^{2}u'^{2}dt^{2}-r^{2}dr'^{2}-d\mathrm {V} '^{2}\right)+2p''r'dr'd\mathrm {V} '-p''^{2}u'^{2}dt^{2}\,;$

et, cette valeur de $\delta ^{2}$ étant substituée dans l’autre équation, on aura

$r'^{2}\left(r^{2}u'^{2}dt^{2}-r^{2}dr'^{2}-d\mathrm {V} '^{2}\right)u^{2}dt^{2}+\left(2p''r'dr'd\mathrm {V} '-p''^{2}u'^{2}dt^{2}\right)u^{2}dt^{2}$

${\begin{aligned}=&(rdr)^{2}\left(r'^{2}u'^{2}dt^{2}-r'^{2}dr'^{2}\right)+d\mathrm {V} ^{2}\left(r^{2}u'^{2}dt^{2}-d\mathrm {V} '^{2}\right)\\&+\left(v''dt^{2}\right)^{2}\left(r^{2}r'^{2}-p''^{2}\right)+2rdrd\mathrm {V} \left(r'dr'd\mathrm {V} '-p''u'^{2}dt^{2}\right)\\&+2rdrv''dt^{2}\left(p''r'dr'-r'^{2}d\mathrm {V} '\right)+2d\mathrm {V} v''dt^{2}\left(p''d\mathrm {V} '-r^{2}r'dr'\right)\,;\end{aligned}}$

ou bien, en ordonnant les termes,

${\begin{aligned}&\left(r^{2}r'^{2}-p''^{2}\right)\left(u^{2}u'^{2}-v''^{2}\right)dt^{4}+(rdr.r'dr'-d\mathrm {V} d\mathrm {V} ')^{2}\\&\quad -\left[r^{2}(r'dr')^{2}-2p''r'dr'd\mathrm {V} '+r'^{2}d\mathrm {V} '^{2}\right]u^{2}dt^{2}\\&\quad -\left[r'^{2}(rdr)^{2}-2p''rdrd\mathrm {V} +r^{2}d\mathrm {V} ^{2}\right]u'^{2}dt^{2}\\&\quad -2\left[p''(rdr.r'dr'+d\mathrm {V} d\mathrm {V} ')-r^{2}r'dr'd\mathrm {V} -r'^{2}rdrd\mathrm {V} '\right]v''dt^{2}=0.\end{aligned}}$

Or (Article V)

$d\mathrm {V} ={\frac {dp''+d\rho }{2}},\quad {\text{et}}\quad d\mathrm {V} '={\frac {dp''-d\rho }{2}}\,;$