Page:Mémoires de l’Académie des sciences, Tome 22.djvu/247

(74)

$\left\{{\begin{aligned}&\qquad \psi _{0}(x,y\ldots )+\psi _{1}(x,y\ldots )+\ldots +\psi _{m-1}(x,y\ldots )=0,\\&\varphi _{0}(\alpha ,\beta \ldots )\psi _{0}(x,y\ldots )+\varphi _{1}(\alpha ,\beta \ldots )\psi _{1}(x,y\ldots )+\ldots \\&\qquad \qquad \qquad \qquad +\varphi _{m-1}(\alpha ,\beta \ldots )\psi _{m-1}(x,y\ldots )=0,\\&\left[\varphi _{0}(\alpha ,\beta ,\gamma \ldots )\right]^{2}\psi _{0}(x,y\ldots )+\left[\varphi _{1}(\alpha ,\beta \ldots )\right]^{2}\psi _{1}(x,y\ldots )+\ldots \\&\qquad \qquad \qquad \qquad +\left[\varphi _{m-1}(\alpha ,\beta \ldots )\right]^{2}\psi _{m-1}(x,y\ldots )=0,\\&{\text{etc}}\ldots \\&\left[\varphi _{0}(\alpha ,\beta ,\gamma \ldots )\right]^{m-1}\psi _{0}(x,y\ldots )+\left[\varphi _{1}(\alpha ,\beta \ldots )\right]^{m-1}\psi _{1}(x,y\ldots )+\ldots \\&\qquad \qquad \qquad \qquad +\left[\varphi _{m-1}(\alpha ,\beta \ldots )\right]^{m-1}\psi _{m-1}(x,y\ldots )=0\end{aligned}}\right.$

Or, en éliminant $\psi _{1}(x,y\ldots ),$ $\psi _{2}(x,y\ldots ),\ldots$ $\psi _{m-1}(x,y\ldots ),$ on tire des équations (74)

(75)

$(\varphi _{0}-\varphi _{1})(\varphi _{0}-\varphi _{2})\ldots (\varphi _{0}-\varphi _{m-1}).\psi _{0}(x,y,z\ldots )=0,$

Soient d’ailleurs

(76)

$\alpha =\chi _{0}(\beta ,\gamma \ldots ),\qquad \alpha =\chi _{1}(\beta ,\gamma \ldots ),\qquad {\text{etc}}\ldots$

les valeurs de $\alpha$ déduites des équations

(77)

$\left\{{\begin{aligned}&\varphi _{0}(\alpha ,\beta ,\gamma \ldots )-\varphi _{1}(\alpha ,\beta ,\gamma \ldots )=0,\\&{\text{etc}}\ldots \\&\varphi _{0}(\alpha ,\beta ,\gamma \ldots )-\varphi _{m-1}(\alpha ,\beta ,\gamma \ldots )=0,\end{aligned}}\right.$

on trouvera pour la valeur générale de $\psi _{0}(x,y,z\ldots )$ déduite de l’équation (75)

(78)

$\psi _{0}(x,y,z\ldots )=e^{x\chi _{0}(\beta ,\gamma \ldots )}{\text{Ȣ}}_{0}(y,z\ldots )+e^{x\chi _{1}(\beta ,\gamma \ldots )}{\text{Ȣ}}_{1}(y,z\ldots )+{\text{etc}}.$

${\text{Ȣ}}_{0},\,{\text{Ȣ}}_{1},\ldots$ indiquant de nouvelles fonctions arbitraires. Des valeurs correspondantes, mais particulières, de $\psi _{1}(x,y,z\ldots ),$ $\psi _{2}(x,y,z\ldots ),\ldots$ tirées des équations (74), seront

(79)

$\left\{{\begin{aligned}&\psi _{1}(x,y,z\ldots )=-{\frac {(\varphi _{0}-\varphi _{2})(\varphi _{0}-\varphi _{3})\ldots (\varphi _{0}-\varphi _{m-1})}{(\varphi _{1}-\varphi _{2})(\varphi _{1}-\varphi _{3})\ldots (\varphi _{1}-\varphi _{m-1})}}\psi _{0}(x,y,z\ldots ),\\&\psi _{2}(x,y,z\ldots )=-{\frac {(\varphi _{0}-\varphi _{1})(\varphi _{0}-\varphi _{3})\ldots (\varphi _{0}-\varphi _{m-1})}{(\varphi _{2}-\varphi _{1})(\varphi _{2}-\varphi _{3})\ldots (\varphi _{2}-\varphi _{m-1})}}\psi _{0}(x,y,z\ldots ),\\&{\text{etc}}\ldots \\&\psi _{m-1}(x,y,z\ldots )=\\&\qquad \qquad -{\frac {(\varphi _{0}-\varphi _{1})(\varphi _{0}-\varphi _{3})\ldots (\varphi _{0}-\varphi _{m-1})}{(\varphi _{m-1}-\varphi _{1})(\varphi _{m-1}-\varphi _{3})\ldots (\varphi _{m-1}-\varphi _{m-1})}}\psi _{0}(x,y,z\ldots ),\end{aligned}}\right.$

et la valeur correspondante de $u$ donnée par la formule (52) deviendra