Page:Lagrange - Œuvres (1867) vol. 1.djvu/332

variables $\mathrm {Y}$ et $\mathrm {Z}$ qui croissent en même temps des quantités très-petites $qt,rt,$ on trouvera par un calcul assez simple

${\begin{aligned}&\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}=\mathrm {L} +t\left(p{\frac {d\mathrm {L} }{d\mathrm {X} }}+q{\frac {d\mathrm {L} }{d\mathrm {Y} }}+r{\frac {d\mathrm {L} }{d\mathrm {Z} }}\right)\\&+{\frac {1}{2}}t^{2}\left(p^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {X} ^{2}}}+2pq{\frac {d^{2}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} }}+q^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Y} ^{2}}}+2pr{\frac {d^{2}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} }}+2qr{\frac {d^{2}\mathrm {L} }{d\mathrm {Y} d\mathrm {Z} }}+r^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Z} ^{2}}}\right)\\&+{\frac {1}{2.3}}t^{3}\left(p^{3}{\frac {d^{3}\mathrm {L} }{d\mathrm {X} ^{3}}}+3p^{2}q{\frac {d^{3}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Y} }}+3pq^{2}{\frac {d^{3}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{2}}}+q^{3}{\frac {d^{3}\mathrm {L} }{d\mathrm {Y} ^{3}}}+3p^{2}r{\frac {d^{3}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Z} }}\right.\\&+\left.3pr^{2}{\frac {d^{3}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} ^{2}}}+3q^{2}r{\frac {d^{3}\mathrm {L} }{d\mathrm {Y} ^{2}d\mathrm {Z} }}+3qr^{2}{\frac {d^{3}\mathrm {L} }{d\mathrm {Y} d\mathrm {Z} ^{2}}}+6pqr{\frac {d^{3}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} }}+r^{3}{\frac {d^{3}\mathrm {L} }{d\mathrm {Z} ^{3}}}\right)\\&+{\frac {1}{2.3.4}}t^{4}\left(p^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{4}}}+4p^{3}q{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{3}d\mathrm {Y} }}+6p^{2}q^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Y} ^{2}}}+4pq^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{3}}}\right.\\&+q^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{4}}}+4p^{3}r{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{3}d\mathrm {Z} }}+6p^{2}r^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Z} ^{2}}}+4pr^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Z} ^{3}}}\\&+4q^{3}r{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{3}d\mathrm {Z} }}+6q^{2}r^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{2}d\mathrm {Z} ^{2}}}+4qr^{3}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} d\mathrm {Z} ^{3}}}+12p^{2}qr{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Y} d\mathrm {Z} }}\\&+\left.12pq^{2}r{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} ^{2}d\mathrm {Z} }}+12pqr^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} d\mathrm {Y} d\mathrm {Z} ^{2}}}+r^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Z} ^{4}}}\right).\end{aligned}}$

De cette formule je déduis les suivantes :

${\begin{aligned}&\quad \ {\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} +pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} +rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} +rt)}\\&+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} +qt,\mathrm {Z} -rt)}+{\frac {1}{8}}\mathrm {L} ^{(\mathrm {X} -pt,\mathrm {Y} -qt,\mathrm {Z} -rt)}\\=&\mathrm {L} +{\frac {1}{2}}t^{2}\left(p^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {X} ^{2}}}+q^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Y} ^{2}}}+r^{2}{\frac {d^{2}\mathrm {L} }{d\mathrm {Z} ^{2}}}\right)\\&+{\frac {1}{2.3.4}}t^{4}\left(p^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{4}}}+6p^{2}q^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Y} ^{2}}}+q^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{4}}}+6p^{2}r^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {X} ^{2}d\mathrm {Z} ^{2}}}\right.\\\end{aligned}}$

$+\left.6q^{2}r^{2}{\frac {d^{4}\mathrm {L} }{d\mathrm {Y} ^{2}d\mathrm {Z} ^{2}}}+r^{4}{\frac {d^{4}\mathrm {L} }{d\mathrm {Z} ^{4}}}\right),$