Page:Annales de mathématiques pures et appliquées, 1830-1831, Tome 21.djvu/282

 Les corrections sont expliquées en page de discussion

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

on trouvera

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

et ainsi de suite ; ce qui donnera finalement

${\cfrac {a+a'x+a''x^{2}+a'''x^{3}+\ldots }{b+b'x+b''x^{2}+b'''x^{3}+\ldots }}={\cfrac {a}{b+{\cfrac {x}{{\cfrac {b}{c}}+{\cfrac {x}{{\cfrac {c}{d}}+{\cfrac {x}{{\cfrac {d}{e}}+{\cfrac {x}{\cfrac {e}{f+\ldots }}}}}}}}}}}\,;$

et, par suite,

${\cfrac {b+b'x+b''x^{2}+b'''x^{3}+\ldots }{a+a'x+a''x^{2}+a'''x^{3}+\ldots }}={\cfrac {x}{{\cfrac {a}{b}}+{\cfrac {x}{{\cfrac {b}{c}}+{\cfrac {x}{{\cfrac {c}{d}}+{\cfrac {x}{{\cfrac {d}{e}}+{\cfrac {x}{\cfrac {e}{f+\ldots }}}}}}}}}}}\,;$

ce qui revient encore à

$ax.{\cfrac {b+b'x+b''x^{2}+b'''x^{3}+\ldots }{a+a'x+a''x^{2}+a'''x^{3}+\ldots }}={\cfrac {abx}{a+{\cfrac {bcx}{b+{\cfrac {cdx}{c+{\cfrac {dex}{d+{\cfrac {efx}{e+\ldots }}}}}}}}}}$