Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/131

Donc, substituant pour ${\frac {1}{\mathrm {Z} }}{\frac {d\mathrm {Z} }{dz}},{\frac {d\mathrm {Z} }{dz}},\mathrm {Z} {\frac {d\mathrm {Z} }{dz}},\ldots .$ les valeurs trouvées dans le même Article, et faisant les réductions enseignées dans l’Article VIII on aura

${\begin{aligned}x=&t-{\frac {n\mathrm {A} '}{2{\sqrt {-1}}}}\left(e^{t{\sqrt {-1}}}-e^{-t{\sqrt {-1}}}\right)+{\frac {n^{2}\mathrm {A} ''}{2^{2}{\sqrt {-1}}}}\left(e^{2t{\sqrt {-1}}}-e^{-2t{\sqrt {-1}}}\right)\\&-{\frac {n^{3}\mathrm {A} '''}{2^{3}{\sqrt {-1}}}}\left(e^{3t{\sqrt {-1}}}-e^{-3t{\sqrt {-1}}}\right)+\ldots ,\end{aligned}}$

ou bien

$x=t-2\left({\frac {n}{2}}\right)\mathrm {A} '\sin t+2\left({\frac {n}{2}}\right)^{2}\mathrm {A} ''\sin 2t-2\left({\frac {n}{2}}\right)^{3}\mathrm {A} '''\sin 3t$

$+2\left({\frac {n}{2}}\right)^{4}\mathrm {A} ^{\scriptscriptstyle {\text{IV}}}\sin 4t-\ldots ,$

où les coefficients $\mathrm {A',A''} ,\ldots$ seront déterminés ainsi

${\begin{aligned}\mathrm {A} '\ \ &=\ \ 1\ \ -3\left({\frac {n}{2}}\right)^{2}\ \ \ {\frac {1}{2.3}}\ \ \ +{\frac {4.5}{2}}\left({\frac {n}{2}}\right)^{4}{\frac {1}{2.3.4.5}}\ \ -\ldots ,\\\mathrm {A} ''\ &=\ {\frac {2}{2}}\ \ -4\left({\frac {n}{2}}\right)^{2}\ \ {\frac {2^{3}}{2.3.4}}\ +{\frac {5.6}{2}}\left({\frac {n}{2}}\right)^{4}{\frac {2^{5}}{2.3\ldots 6}}-\ldots ,\\\mathrm {A} '''&={\frac {3^{2}}{2.3}}-5\left({\frac {n}{2}}\right)^{2}{\frac {3^{4}}{2.3.4.5}}+{\frac {6.7}{2}}\left({\frac {n}{2}}\right)^{4}{\frac {3^{6}}{2.3\ldots 7}}-\ldots ,\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

c’est-à-dire, en faisant ${\frac {n}{2}}=\nu ,$

${\begin{aligned}\mathrm {A} '\ \ &=1-{\frac {\nu ^{2}}{2}}+{\frac {\nu ^{4}}{2^{2}.3}}-{\frac {\nu ^{6}}{2^{2}.3^{2}.4}}+\ldots ,\\\mathrm {A} ''\ &={\frac {2}{2}}-{\frac {2^{3}\nu ^{2}}{2.3}}+{\frac {2^{5}\nu ^{4}}{2^{2}.3.4}}-{\frac {2^{7}\nu ^{6}}{2^{2}.3^{2}.4.5}}+\ldots \\\mathrm {A} '''&={\frac {3^{2}}{2.3}}-{\frac {3^{4}\nu ^{2}}{2.3.4}}+{\frac {3^{6}\nu ^{4}}{2^{2}.3.4.5}}-{\frac {3^{8}\nu ^{6}}{2^{2}.3^{2}.4.5.6}}+\ldots \\\mathrm {A} ^{\scriptscriptstyle {\text{IV}}}&={\frac {4^{3}}{2.3.4}}-{\frac {4^{5}\nu ^{2}}{2.3.4.5}}+{\frac {4^{7}\nu ^{4}}{2^{2}.3.4.5.6}}-{\frac {4^{9}\nu ^{6}}{2^{2}.3^{2}.4.5.6.7}}+\ldots \\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$