Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/300

Donc, faisant ici les mêmes substitutions que ci-dessus, et supposant, pour abréger,

$(k)\qquad \qquad \left\{{\begin{aligned}\mathrm {L} \ =&{\frac {d^{2}(r^{2})}{2dt^{2}}}\,+{\frac {\mathrm {A+B} }{r}}-{\frac {\mathrm {D} \left(3z^{2}-r^{2}\right)}{\mathrm {R} ^{3}}},\\\mathrm {M} =&{\frac {d^{2}(\mathrm {R} ^{2})}{2dt^{2}}}+\mathrm {\frac {D}{R}} ,\\\mathrm {N} \,=&{\frac {d^{2}(\mathrm {R} z)}{dt^{2}}}+{\frac {\mathrm {(A+B)R} z}{r^{3}}}-{\frac {\mathrm {D} z}{\mathrm {R} ^{2}}},\end{aligned}}\right.$

on aura

${\begin{aligned}u^{2}\ \,=&\mathrm {L} -{\frac {i\mathrm {D} \left(15z^{3}-9r^{2}z\right)}{2\mathrm {R} ^{4}}}-\ldots ,\\u'^{2}\,=&{\frac {\mathrm {M} }{i^{2}}}+{\frac {\mathrm {BR} z}{ir^{3}}}+\ldots ,\\u'^{2}=&{\frac {\mathrm {M} }{i^{2}}}+{\frac {1}{i}}\left({\frac {\mathrm {BR} z}{r^{3}}}-\mathrm {N} \right)+\left[\mathrm {L} -{\frac {\mathrm {B} }{r}}+{\frac {3\mathrm {D} \left(3z^{2}-r^{2}\right)}{2\mathrm {R} ^{3}}}\right]+\ldots .\end{aligned}}$

Or on a

$\mathrm {P} =pp'+pp''+p'p''=pr^{2}+p'p''={\frac {\mathrm {R} ^{2}\left(r^{2}-z^{2}\right)}{i^{2}}}\,;$

donc

${\begin{aligned}\mathrm {P} u^{2}\ \,=&{\frac {\mathrm {L} \mathrm {R} ^{2}\left(r^{2}-z^{2}\right)}{i^{2}}}-{\frac {\mathrm {D} \left(r^{2}-z^{2}\right)\left(15z^{3}-9r^{2}z\right)}{2i\mathrm {R} ^{2}}}-\ldots ,\\\mathrm {P} u'^{2}\,=&{\frac {\mathrm {M} \mathrm {R} ^{2}\left(r^{2}-z^{2}\right)}{i^{4}}}+{\frac {\mathrm {BR^{3}} z\left(r^{2}-z^{2}\right)}{i^{3}r^{3}}}+\ldots ,\\\mathrm {P} u'^{2}=&{\frac {\mathrm {M} \mathrm {R} ^{2}\left(r^{2}-z^{2}\right)}{i^{4}}}+{\frac {\mathrm {BR^{3}} z\left(r^{2}-z^{2}\right)}{i^{3}r^{3}}}-{\frac {\mathrm {NR^{2}} \left(r^{2}-z^{2}\right)}{i^{3}}}-\ldots .\end{aligned}}$

De plus, en faisant, pour abréger,

${\begin{aligned}\Gamma =&{\frac {\mathrm {R} ^{2}\left[d\left(r^{2}\right)\right]^{2}+r^{2}\left[d(\mathrm {R} z)\right]^{2}-2\mathrm {R} zd(\mathrm {R} z)d\left(r^{2}\right)}{dt^{2}}}\\&\ \ \quad \qquad \qquad \qquad \qquad \qquad +{\frac {2\sigma \left[\mathrm {R} zd\left(r^{2}\right)-r^{2}d(\mathrm {R} z)\right]}{dt}}+r^{2}\sigma ^{2},\\\Delta =&{\frac {\mathrm {R} ^{2}\left[d\left(\mathrm {R} z\right)\right]^{2}+r^{2}\left[d\left(\mathrm {R} ^{2}\right)\right]^{2}-2\mathrm {R} zd(\mathrm {R} z)d\left(\mathrm {R} ^{2}\right)}{dt^{2}}}\\&\quad \qquad \qquad \qquad \qquad \qquad +{\frac {2\sigma \left[\mathrm {R} ^{2}d(\mathrm {R} z)-\mathrm {R} zd\left(\mathrm {R} ^{2}\right)\right]}{dt}}+\mathrm {R} ^{2}\sigma ^{2},\\\Lambda =&{\frac {\mathrm {R} z\left[d\left(\mathrm {R} ^{2}\right)d\left(r^{2}\right)+[d(\mathrm {R} z)]^{2}\right]-d(\mathrm {R} z)d\left(\mathrm {R} ^{2}r^{2}\right)}{dt^{2}}}\\&\qquad \qquad \qquad \qquad \qquad \qquad -{\frac {\sigma \left[\mathrm {R} ^{2}d\left(r^{2}\right)-r^{2}d\left(\mathrm {R} ^{2}\right)\right]}{dt}}-\mathrm {R} z\sigma ^{2},\end{aligned}}$