Page:Lagrange - Œuvres (1867) vol. 1.djvu/643

c’est-à-dire

${\begin{aligned}\mathrm {D} =&\gamma +\lambda \delta +i\left(\mu \gamma ^{2}+\mu _{1}\gamma \delta +\mu _{2}\delta ^{2}+\mu _{3}\Gamma \right))\\&+i^{2}\left(\nu \gamma ^{3}+s\nu _{1}\gamma ^{2}\delta +\nu _{2}\gamma \delta ^{2}+\nu _{3}\delta ^{3}+\nu _{4}\Delta +\nu _{5}\Lambda +\nu _{6}\gamma \Gamma +\nu _{7}\delta \Gamma \right)\end{aligned}}$

et

$\mathrm {E} =\varepsilon +\lambda \eta +i\left[2\mu \gamma \varepsilon +\mu _{1}(\delta \varepsilon +\gamma \eta )+2\mu _{2}\delta \eta +\mu _{3}\delta \varepsilon \right]$

$+i\left[3\nu \gamma ^{2}\delta +\nu _{1}\left(2\gamma \delta \varepsilon +\gamma ^{2}\eta \right)+\nu _{2}\left(\delta ^{2}\varepsilon +2\gamma \delta \eta \right)+3\nu _{3}\delta ^{2}\eta +\nu _{4}\gamma \delta \varepsilon +\nu _{5}\delta ^{2}\varepsilon \right.$

$\left.+\nu _{6}\left(\Gamma \varepsilon +\gamma ^{2}\delta \varepsilon \right)+\nu _{7}\left(\Gamma \eta +\gamma \delta ^{2}\varepsilon \right)\right]\,;$

l’équation (Q) donnera, en divisant par $e^{\rho t},$

${\frac {d\theta }{dt}}-\rho \theta +{\frac {\varkappa }{\rho }}=\left(\mathrm {E} -\rho \mathrm {D} +{\frac {\varkappa }{\rho }}\right)e^{-\rho t},$

et prenant la quantité $\rho$ en $-,$

${\frac {d\theta }{dt}}+\rho \theta -{\frac {\varkappa }{\rho }}=\left(\mathrm {E} +\rho \mathrm {D} -{\frac {\varkappa }{\rho }}\right)e^{\rho t},$

d’où l’on tire

$\theta ={\frac {\varkappa }{\rho ^{2}}}+\left(\mathrm {D} -{\frac {\varkappa }{\rho ^{2}}}\right){\frac {e^{\rho t}+e^{-\rho t}}{2}}+\mathrm {E} {\frac {e^{\rho t}-e^{-\rho t}}{2\rho }}$

ou bien

$\theta ={\frac {\varkappa }{\rho ^{2}}}+\left(\mathrm {D} -{\frac {\varkappa }{\rho ^{2}}}\right)\cos \left(t{\sqrt {-\rho ^{2}}}\right)+{\frac {\mathrm {E} }{\sqrt {-\rho ^{2}}}}\sin \left(t{\sqrt {-\rho ^{2}}}\right).$

Soient maintenant $\rho '^{2}$ et $\rho ''^{2}$ les deux racines de l’équation (R), et $\theta ',\theta '',\lambda ',\lambda '',\mu ',\mu '',\ldots$ les valeurs correspondantes de $\theta ,\lambda ,\mu ,\ldots \,;$ on aura, en remettant au lieu de $u,u_{1},u_{2},\ldots ,$ leurs valeurs $y^{2},yz,z^{2},\ldots$ ces deux équations

${\begin{aligned}y&+\lambda 'z+i\left(\mu 'y^{2}+\mu '_{1}yz+\mu '_{2}z^{2}+\mu '_{3}\int zdy\right)\\&+i^{2}\left(\nu 'y^{3}+\nu '_{1}y^{2}z+\nu '_{2}yz^{2}+\nu '_{3}z^{3}+\nu '_{4}\int yzdy\right.\end{aligned}}$

$\left.+\nu '_{5}\int z^{2}dy+\nu '_{6}y\int dzy+\nu '_{7}z\int zdy\right)=\theta ',$