Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/123

de sorte qu’on trouvera, après quelques réductions fort simples,

$u={\frac {t}{m}}-{\frac {2n}{m}}sinnt$

${\begin{aligned}&-{\frac {n^{2}}{2}}\left({\frac {2}{m^{3}}}\int \sin ^{2}tdt-{\frac {4}{2m}}{\frac {d\sin ^{2}t}{dt}}\right)\\&+{\frac {n^{3}}{3}}\left({\frac {4}{m^{3}}}\sin ^{3}tdt-{\frac {6}{2.3.m}}{\frac {d^{2}\sin ^{3}t}{dt^{2}}}\right)\\&+{\frac {n^{4}}{4}}\left({\frac {4}{m^{5}}}\int \sin ^{4}tdt-{\frac {6}{2m^{3}}}{\frac {d\sin ^{4}t}{dt}}+{\frac {8}{2.3.4.m}}{\frac {d^{3}\sin ^{4}t}{dt^{3}}}\right)\\&-{\frac {n^{5}}{5}}\left({\frac {6}{m^{5}}}\sin ^{5}t-{\frac {8}{2.3.m^{3}}}{\frac {d^{2}\sin ^{5}t}{dt^{2}}}+{\frac {10}{2.3.4.5.m}}{\frac {d^{4}\sin ^{5}t}{dt^{4}}}\right)\\&+{\frac {n^{6}}{6}}\left({\frac {6}{m^{7}}}\int \sin ^{6}tdt-{\frac {8}{2.m^{5}}}{\frac {d\sin ^{6}t}{dt}}+{\frac {10}{2.3.4.m^{3}}}{\frac {d^{3}\sin ^{6}t}{dt^{3}}}\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.-{\frac {12}{2.3.4.5.6.m}}{\frac {d^{5}\sin ^{6}t}{dt^{5}}}\right)\\&+{\frac {n^{7}}{7}}\left({\frac {8}{m^{7}}}\sin ^{7}t-{\frac {10}{2.3.m^{5}}}{\frac {d^{2}\sin ^{7}t}{dt^{2}}}+{\frac {12}{2.3.4.5.m^{3}}}{\frac {d^{4}\sin ^{7}t}{dt^{4}}}\right.\end{aligned}}$

$\left.-{\frac {14}{2.3.4.5.6.7.m}}{\frac {d^{6}\sin ^{7}t}{dt^{6}}}\right)$

$\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .$

Réduisant maintenant les puissances de $\sin t$ en sinus et cosinus de multiples de $t$ par les formules de l’Article III et exécutant les intégrations et les différentiations indiquées, on aura

${\begin{aligned}u=&{\frac {t}{m}}-{\frac {2n}{m}}\sin t\\&-{\frac {n^{2}}{2.2}}\left[{\frac {2.2}{2m^{3}}}t-\left({\frac {2}{2m^{3}}}+{\frac {4.2}{2m}}\right)\sin 2t\right]\\&+{\frac {n^{3}}{4.3}}\left[3\left({\frac {4}{m^{3}}}+{\frac {6}{2.3.m}}\right)\sin t-\left({\frac {4}{m^{3}}}+{\frac {6.3^{2}}{2.3.m}}\right)\sin 3t\right]\\&+{\frac {n^{4}}{8.4}}\left[{\frac {4.4.3}{2.2.m^{5}}}t-4\left({\frac {4}{2m^{5}}}+{\frac {6.2}{2m^{3}}}+{\frac {8.2^{3}}{2.3.4.m}}\right)\sin 2t\right.\end{aligned}}$

$\left.+\left({\frac {4}{4m^{5}}}+{\frac {6.4}{2m^{3}}}+{\frac {8.4^{3}}{2.3.4.m}}\right)\sin 4t\right]$