Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/124

quera seulement que, à cause de $1-{\frac {\mu ^{2}}{f}}=ng$ (Article XXIX), on aura, en négligeant la quantité $ng$ qui est de l’ordre $n,$

$f=\mu ^{2},\quad$et de là$\quad f_{1}=\mu _{1}^{2},\quad f_{2}=\mu _{2}^{2},\quad f_{3}=\mu _{3}^{2},\quad f_{4}=\mu _{4}^{2}$

par conséquent (Article XXXV)

$\mathrm {M} _{1}=\mu _{1},\quad$et pareillement$\quad \mathrm {M} _{2}=\mu _{2},\quad \mathrm {M} _{3}=\mu _{3},\quad \mathrm {M} _{4}=\mu _{4}.$

Donc, supposant $s={\frac {\mu _{2}}{\mu _{1}}},$ on aura

${\begin{aligned}&\Xi _{1}(a_{1},a_{2})=\left[{\breve {\Gamma }}_{1}(a_{1},a_{2})+{\frac {2}{s-1}}{\breve {\Gamma }}_{1}(a_{1},a_{2})\right]{\frac {1}{(s-1)^{2}-1}},\\&\Xi _{2}(a_{1},a_{2})=\left[{\breve {\Gamma }}_{2}(a_{1},a_{2})+{\frac {2}{2(s-1)}}{\breve {\Gamma }}_{2}(a_{1},a_{2})\right]{\frac {1}{4(s-1)^{2}-1}},\\&\Xi _{3}(a_{1},a_{2})=\left[{\breve {\Gamma }}_{3}(a_{1},a_{2})+{\frac {2}{3(s-1)}}{\breve {\Gamma }}_{3}(a_{1},a_{2})\right]{\frac {1}{9(s-1)^{2}-1}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\&\Phi _{1}(a_{1},a_{2})={\frac {2}{s-1}}\Xi _{1}(a_{1},a_{2})+{\frac {1}{(s-1)^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{2}),\\&\Phi _{2}(a_{1},a_{2})={\frac {2}{2(s-1)}}\Xi _{2}(a_{1},a_{2})+{\frac {1}{4(s-1)^{2}}}{\widehat {\Gamma }}_{2}(a_{1},a_{2}),\\&\Phi _{3}(a_{1},a_{2})={\frac {2}{3(s-1)}}\Xi _{3}(a_{1},a_{2})+{\frac {1}{9(s-1)^{2}}}{\widehat {\Gamma }}_{3}(a_{1},a_{2}),\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

De méme, en faisant $s={\frac {\mu _{3}}{\mu _{1}}},$ on aura

${\begin{aligned}&\Xi _{1}(a_{1},a_{3})=\left[{\breve {\Gamma }}_{1}(a_{1},a_{3})+{\frac {2}{s-1}}{\widehat {\Gamma }}(a_{1},a_{3})\right]{\frac {1}{(s-1)^{2}-1}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\&\Phi _{1}(a_{1},a_{3})={\frac {2}{s-1}}\Xi _{1}(a_{1},a_{3})+{\frac {1}{(s-1)^{2}}}{\widehat {\Gamma }}_{1}(a_{1},a_{3}),\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$