Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/196

et

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

CV.

On peut simplifier cette expression de $\mathrm {x} _{1}$ en supposant, en général,

$\mathrm {D} _{1}=-\delta _{1}\sin \omega _{1},\quad \mathrm {E} _{1}=-\delta _{1}\cos \omega _{1}\,;$

ce qui donne

$\delta _{1}=\mathrm {\sqrt {D_{1}^{2}+E_{1}^{2}}} ,\quad \operatorname {tang} \omega _{1}=\mathrm {\frac {D_{1}}{E_{1}}} ,$

savoir

$\delta =\mathrm {\sqrt {(Y_{1}+A_{1}Y_{2}+B_{1}Y_{3}+C_{1}Y_{4})^{2}+(\mu _{1}X_{1}+A_{1}\mu _{2}X_{2}+B_{1}\mu _{3}X_{3}+C_{1}\mu _{4}X_{4})^{2}}} ,$

$\operatorname {tang} \omega =-\mathrm {\frac {Y_{1}+A_{1}Y_{2}+B_{1}Y_{3}+C_{1}Y_{4}}{\mu _{1}X_{1}+A_{1}\mu _{2}X_{2}+B_{1}\mu _{3}X_{3}+C_{1}\mu _{4}X_{4}}} .$

Par ce moyen, on aura

$\mathrm {D} _{1}\sin \left(\mu _{1}-{\frac {n}{2}}\rho \right)t-\mathrm {E} _{1}\cos \left(\mu _{1}-{\frac {n}{2}}\rho \right)t=\delta _{1}\cos \left[\left(\mu _{1}-{\frac {n}{2}}\rho \right)t+\omega _{1}\right],$

et de là

${\begin{aligned}\mathrm {D} '_{1}\ \sin \left(\mu _{1}-{\frac {n}{2}}\rho '\right)t-\mathrm {E} '_{1}\ \cos \left(\mu _{1}-{\frac {n}{2}}\rho '\right)t=&\delta '_{1}\,\cos \left[\left(\mu _{1}-{\frac {n}{2}}\rho '\right)t+\omega '_{1}\,\right],\\\mathrm {D} ''_{1}\sin \left(\mu _{1}-{\frac {n}{2}}\rho ''\right)t-\mathrm {E} ''_{1}\cos \left(\mu _{1}-{\frac {n}{2}}\rho ''\right)t=&\delta ''_{1}\cos \left[\left(\mu _{1}-{\frac {n}{2}}\rho ''\right)t+\omega ''_{1}\right]\,;\end{aligned}}$

et ainsi de suite.