Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/206

Or

${\begin{aligned}\sin &\left[\mu _{1}-{\frac {n}{2}}\left(p\pm q{\sqrt {-1}}\right)\right]t\\&=\sin \left(\mu _{1}-{\frac {n}{2}}p\right)t\times \cos {\frac {n}{2}}qt{\sqrt {-1}}\mp \cos \left(\mu _{1}-{\frac {n}{2}}p\right)t\times \sin {\frac {n}{2}}qt{\sqrt {-1}},\end{aligned}}$

et de même

${\begin{aligned}\cos &\left[\mu _{1}-{\frac {n}{2}}\left(p\pm q{\sqrt {-1}}\right)\right]t\\&=\cos \left(\mu _{1}-{\frac {n}{2}}p\right)t\times \cos {\frac {n}{2}}qt{\sqrt {-1}}\pm \sin \left(\mu _{1}-{\frac {n}{2}}p\right)t\times \sin {\frac {n}{2}}qt{\sqrt {-1}}.\end{aligned}}$

Donc les termes

${\begin{alignedat}{2}{\frac {\beta _{1}\mathrm {D} '''_{1}}{\mu _{1}(1+\alpha _{1}+\beta _{1}+\gamma _{1})}}\sin \left(\mu _{1}-{\frac {n}{2}}\rho '''\right)t&-{\frac {\beta _{1}\mathrm {E} '''_{1}}{\mu _{1}(1+\alpha _{1}+\beta _{1}+\gamma _{1})}}\cos \left(\mu _{1}-{\frac {n}{2}}\rho '''\right)t\\{\frac {\gamma _{1}\mathrm {D} _{1}^{\scriptscriptstyle {\text{IV}}}}{\mu _{1}(1+\alpha _{1}+\beta _{1}+\gamma _{1})}}\sin \left(\mu _{1}-{\frac {n}{2}}\rho ^{\scriptscriptstyle {\text{IV}}}\right)t&-{\frac {\gamma _{1}\mathrm {E} _{1}^{\scriptscriptstyle {\text{IV}}}}{\mu _{1}(1+\alpha _{1}+\beta _{1}+\gamma _{1})}}\cos \left(\mu _{1}-{\frac {n}{2}}\rho ^{\scriptscriptstyle {\text{IV}}}\right)t\end{alignedat}}$

de la valeur de $\mathrm {x} _{1}$ (Article CIV) se changeront en

${\begin{aligned}2&\left[\mathrm {H} \sin \left(\mu _{1}-{\frac {n}{2}}p\right)t-h\cos \left(\mu _{1}-{\frac {n}{2}}p\right)t\right]\cos {\frac {n}{2}}qt{\sqrt {-1}}\\-2&\left[\mathrm {L} \cos \left(\mu _{1}-{\frac {n}{2}}p\right)t+l\sin \left(\mu _{1}-{\frac {n}{2}}p\right)t\right]\sin {\frac {n}{2}}qt{\sqrt {-1}}\times {\sqrt {-1}},\end{aligned}}$

c’est-à-dire, en mettant au lieu de $\cos {\frac {n}{2}}qt{\sqrt {-1}}$ et $\sin {\frac {n}{2}}qt{\sqrt {-1}}$ leurs valeurs exponentielles ${\frac {e^{-{\frac {n}{2}}qt}+e^{{\frac {n}{2}}qt}}{2}},\ {\frac {e^{-{\frac {n}{2}}qt}-e^{{\frac {n}{2}}qt}}{2}},$

${\begin{aligned}e^{-{\frac {n}{2}}qt}&\left[(\mathrm {H} -l)\sin \left(\mu _{1}-{\frac {n}{2}}p\right)t+(\mathrm {L} +h))\cos \left(\mu _{1}-{\frac {n}{2}}p\right)t\right]\\+e^{-qt}&\left[(\mathrm {H} +l)\sin \left(\mu _{1}-{\frac {n}{2}}p\right)t-(\mathrm {L} -h))\cos \left(\mu _{1}-{\frac {n}{2}}p\right)t\right],\end{aligned}}$