Page:Lagrange - Œuvres (1867) vol. 1.djvu/330

Or les équations (A), (B), (C) donnent d’abord

${\begin{aligned}k\mathrm {L} \ =&{\frac {d^{2}\mathrm {L} }{d\mathrm {X} ^{2}}}+{\frac {d^{2}\mathrm {M} }{d\mathrm {X} d\mathrm {Y} }}+{\frac {d^{2}\mathrm {N} }{d\mathrm {X} d\mathrm {Z} }},\\k\mathrm {M} =&{\frac {d^{2}\mathrm {M} }{d\mathrm {Y} ^{2}}}+{\frac {d^{2}\mathrm {L} }{d\mathrm {Y} d\mathrm {X} }}+{\frac {d^{2}\mathrm {N} }{d\mathrm {Y} d\mathrm {Z} }},\\k\mathrm {N} \,=&{\frac {d^{2}\mathrm {N} }{d\mathrm {Z} ^{2}}}+{\frac {d^{2}\mathrm {L} }{d\mathrm {Z} d\mathrm {X} }}+{\frac {d^{2}\mathrm {M} }{d\mathrm {Z} d\mathrm {Y} }}\,;\end{aligned}}$

multipliant ces équations par $k$ et substituant de nouveau au lieu de $k\mathrm {L} ,k\mathrm {M} ,k\mathrm {N}$ leurs valeurs, on aura ensuite

${\begin{aligned}k^{2}\mathrm {L} =&{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{4}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Y} ^{2}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Z} ^{2}}}+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} ^{3}d\mathrm {Y} }}+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} d\mathrm {Y} ^{3}}}+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} ^{2}}}\\&+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} ^{3}d\mathrm {Z} }}+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} d\mathrm {Z} ^{3}}}+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} d\mathrm {Z} d\mathrm {Y} ^{2}}},\\k^{2}\mathrm {M} =&{\frac {d^{4}\mathrm {M} }{d\mathrm {Y} ^{4}}}+{\frac {d^{4}\mathrm {M} }{d\mathrm {Y} ^{2}d\mathrm {X} ^{2}}}+{\frac {d^{4}\mathrm {M} }{d\mathrm {Y} ^{2}d\mathrm {Z} ^{2}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{3}d\mathrm {X} }}+{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} d\mathrm {X} ^{3}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} ^{2}}}\\&+{\frac {d^{4}\mathrm {N} }{d\mathrm {Y} ^{3}d\mathrm {Z} }}+{\frac {d^{4}\mathrm {N} }{d\mathrm {Y} d\mathrm {Z} ^{3}}}+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} ^{2}d\mathrm {Y} d\mathrm {Z} }},\\k^{2}\mathrm {N} =&{\frac {d^{4}\mathrm {N} }{d\mathrm {Z} ^{4}}}+{\frac {d^{4}\mathrm {N} }{d\mathrm {Z} ^{2}d\mathrm {X} ^{2}}}+{\frac {d^{4}\mathrm {N} }{d\mathrm {Z} ^{2}d\mathrm {Y} ^{2}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {Z} ^{3}d\mathrm {X} }}+{\frac {d^{4}\mathrm {L} }{d\mathrm {Z} d\mathrm {X} ^{3}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{2}d\mathrm {Z} }}\\&+{\frac {d^{4}\mathrm {M} }{d\mathrm {Z} ^{3}d\mathrm {Y} }}+{\frac {d^{4}\mathrm {M} }{d\mathrm {Z} d\mathrm {Y} ^{3}}}+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} ^{2}d\mathrm {Y} d\mathrm {Z} }}.\end{aligned}}$

Par là, on pourra faire évanouir la lettre $k$ de l’expression précédente ; car on aura

(F)

$\left\{{\begin{aligned}\mathrm {L} +&{\frac {1}{2}}t^{2}ck\mathrm {L} +{\frac {1}{2.3.4}}t^{4}c^{2}k^{2}\mathrm {L} \\=&\mathrm {L} +{\frac {1}{2}}t^{2}c\left({\frac {d^{2}\mathrm {L} }{d\mathrm {X} ^{2}}}+{\frac {d^{2}\mathrm {M} }{d\mathrm {X} d\mathrm {Y} }}+{\frac {d^{2}\mathrm {N} }{d\mathrm {X} d\mathrm {Z} }}\right)\\&+{\frac {1}{2.3.4}}t^{4}c^{2}\left({\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{4}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Y} ^{2}}}+{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Z} ^{2}}}+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} ^{3}d\mathrm {Y} }}+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} d\mathrm {Y} ^{3}}}\right.\\&\ \ \qquad \qquad \qquad \left.+{\frac {d^{4}\mathrm {M} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} ^{2}}}+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} ^{3}d\mathrm {Z} }}+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} d\mathrm {Z} ^{3}}}+{\frac {d^{4}\mathrm {N} }{d\mathrm {X} d\mathrm {Y} ^{2}d\mathrm {Z} }}\right).\end{aligned}}\right.$