Page:Galois - Œuvres mathématiques, Gauthier-Villars, 1897.djvu/36

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positive de $y$ est

$y=1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\;\ldots }}}}}}}}}}}\,;$

sa valeur négative sera donc

$y=-{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ldots }}}}}}}}}}}}\,;$

les deux valeurs de $x$ seront donc

$x=3+{\frac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ldots }}}}}}}}}}}},\quad x=3-{\frac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ldots }}}}}}}}}}}}\,;$

dont la dernière, en vertu de la formule connue

$p-{\frac {1}{q}}=p-1+{\frac {1}{1+{\cfrac {1}{q-1}}}},$

devient

$x=1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+{\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{1+\ldots }}}}}}}}}}}}}}}}}}.$

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