Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/11

De là il s’ensuit qu’on aura

${\begin{aligned}\mathrm {M} =&{\frac {1}{m}}\left({\frac {b}{a}}\right)^{m}\\&+\alpha \left({\frac {b}{a}}\right)^{m-2}+\alpha _{1}\left({\frac {b}{a}}\right)^{m-3}+\ldots +\alpha _{m-2}\\&+{\frac {1}{2}}\left[(m-3)\beta \left({\frac {b}{a}}\right)^{m-4}+(m-4)\beta _{1}\left({\frac {b}{a}}\right)^{m-5}+\ldots +\beta _{m-4}\right]\\&+{\frac {1}{2.3}}{\Biggl [}(m-4)(m-5)\gamma \left({\frac {b}{a}}\right)^{m-6}+(m-5)(m-6)\gamma _{1}\left({\frac {b}{a}}\right)^{m-7}{\Biggr .}\\&\qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\Biggl .}+\ldots +1.2.\gamma _{m-6}{\Biggr ]}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Donc, puisque

$\mathrm {M} m={\frac {1}{p^{m}}}+{\frac {1}{q^{m}}}+{\frac {1}{r^{m}}}+\ldots ,$

on aura, en général,

${\frac {1}{p^{m}}}+{\frac {1}{q^{m}}}+{\frac {1}{r^{m}}}+\ldots$

${\begin{aligned}=&\left({\frac {b}{a}}\right)^{m}+m\alpha \left({\frac {b}{a}}\right)^{m-2}+m\alpha _{1}\left({\frac {b}{a}}\right)^{m-3}+\ldots +m\alpha _{m-2}\\&+{\frac {m(m-3)}{2}}\beta \left({\frac {b}{a}}\right)^{m-4}+{\frac {m(m-4)}{2}}\beta _{1}\left({\frac {b}{a}}\right)^{m-5}+\ldots +{\frac {m}{2}}\beta _{m-4}\\&+{\frac {m(m-4)(m-5)}{2.3}}\gamma \left({\frac {b}{a}}\right)^{m-6}+{\frac {m(m-5)(m-6)}{2.3}}\gamma _{1}\left({\frac {b}{a}}\right)^{m-7}\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +\ldots +{\frac {m}{3}}\gamma _{m-6}\\&+{\frac {m(m-5)(m-6)(m-7)}{2.3.4}}\delta \left({\frac {b}{a}}\right)^{m-8}\\&\quad \qquad \qquad +{\frac {m(m-6)(m-7)(m-8)}{2.3.4}}\delta _{1}\left({\frac {b}{a}}\right)^{m-9}+\ldots +{\frac {m}{4}}\gamma _{m-8}\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

4. Exemple I. — Soit l’équation du second degré

$a-bx+cx^{2}=0\,;$

on aura, dans ce cas,

$\mathrm {X} =-{\frac {cx^{2}}{a}}\,;$

donc

$\mathrm {X} ^{2}={\frac {c^{2}x^{4}}{a^{2}}},\quad \mathrm {X} ^{3}=-{\frac {c^{3}x^{6}}{a^{3}}},\ldots ,$