Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/214

${\begin{aligned}&\operatorname {F} (y_{1})R_{1}=t_{0}+t_{1}y_{1}+t_{2}y_{1}^{2}+t_{3}y_{1}^{3}+\ldots +t_{n-1}y_{1}^{n-1},\\\\&\operatorname {F} (y_{2})R_{2}=t_{0}+t_{1}y_{2}+t_{2}y_{2}^{2}+t_{3}y_{2}^{3}+\ldots +t_{n-1}y_{2}^{n-1},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&\operatorname {F} (y_{n-1})R_{n-1}=t_{0}+t_{1}y_{n-1}+t_{2}y_{n-1}^{2}+t_{3}y_{n-1}^{3}+\ldots +t_{n-1}y_{n-1}^{n-1}.\end{aligned}}$

Maintenant, je dès qu’on aura

$\operatorname {F} (y)={\frac {t_{n-1}}{\rho _{n-1}}}.$

En effet, on a d’abord

$\Sigma {\frac {R}{\psi '(y)}}={\frac {R}{\psi '(y)}}+{\frac {R_{1}}{\psi '(y_{1})}}+{\frac {R_{2}}{\psi '(y_{2})}}+\ldots +{\frac {R_{n-1}}{\psi '(y_{n-1})}}\,;$

donc, en substituant les valeurs de $R,R_{1},R_{2},\ldots ,R_{n-1},$

${\begin{aligned}\Sigma {\frac {R}{\psi '(y)}}&=\rho _{0}\left\{{\frac {1}{\psi '(y)}}+{\frac {1}{\psi '(y_{1})}}+{\frac {1}{\psi '(y_{2})}}+\ldots +{\frac {1}{\psi '(y_{n-1})}}\right\}\\\\&+\rho _{1}\left\{{\frac {y}{\psi '(y)}}+{\frac {y_{1}}{\psi '(y_{1})}}+{\frac {y_{2}}{\psi '(y_{2})}}+\ldots +{\frac {y_{n-1}}{\psi '(y_{n-1})}}\right\}\\\\&+\rho _{2}\left\{{\frac {y^{2}}{\psi '(y)}}+{\frac {y_{1}^{2}}{\psi '(y_{1})}}+{\frac {y_{2}^{2}}{\psi '(y_{2})}}+\ldots +{\frac {y_{n-1}^{2}}{\psi '(y_{n-1})}}\right\}\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+\rho _{n-1}\left\{{\frac {y^{n-1}}{\psi '(y)}}+{\frac {y_{1}^{n-1}}{\psi '(y_{1})}}+{\frac {y_{2}^{n-1}}{\psi '(y_{2})}}+\ldots +{\frac {y_{n-1}^{n-1}}{\psi '(y_{n-2})}}\right\}.\end{aligned}}$

Or, $y,y_{1},y_{2},\ldots y_{n-1},$ étant les racines de l’équation (2) on a