Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/281

 Les corrections sont expliquées en page de discussion

${\begin{alignedat}{1}{\frac {ba'-ab'}{b}}&=c,\\\\{\frac {ba''-ab''}{b}}&=c',\\\\{\frac {ba'''-ab'''}{b}}&=c'',\\\ldots \ldots &\ldots ,\end{alignedat}}$

on aura

${\frac {a+a'x+a''x^{2}+a'''x^{3}+\ldots }{b+b'x+b''x^{2}+b'''x^{3}+\ldots }}={\frac {a}{b}}+{\frac {x}{\cfrac {b+b'x+b''x^{2}+b'''x^{3}+\ldots }{c+c'x+c''x^{2}+c'''x^{3}+\ldots }}}.$

Par un semblable calcul, en posant, pour abréger,

${\begin{aligned}{\frac {cb'-bc'}{c}}\ =&d,\\\\{\frac {cb''-bc''}{c}}\,=&d',\\\\{\frac {cb'''-bc'''}{c}}=&d'',\\\ldots \ldots &\ldots ,\end{aligned}}$

on aura

${\frac {b+b'x+b''x^{2}+b'''x^{3}+\ldots }{c+c'x+c''x^{2}+c'''x^{3}+\ldots }}={\frac {b}{c}}+{\frac {x}{\cfrac {c+c'x+c''x^{2}+c'''x^{3}+\ldots }{d+d'x+d''x^{2}+d'''x^{3}+\ldots }}}.$

En posant semblablement

${\begin{alignedat}{1}{\frac {dc'-cd'}{d}}&=e,\\\ldots \ldots &\ldots ,\end{alignedat}}$