Page:Annales de mathématiques pures et appliquées, 1818-1819, Tome 9.djvu/80

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DES DÉVELOPPANTES

${\begin{aligned}S_{1}=&\phi S_{0}-\int \phi \operatorname {d} S_{0},\\S_{2}=&{\frac {\phi ^{2}}{2!}}S_{0}-{\frac {\phi }{1!}}\int \phi \operatorname {d} S_{0}+\int {\frac {\phi ^{2}}{2!}}\operatorname {d} S_{0},\\S_{3}=&{\frac {\phi ^{3}}{3!}}S_{0}-{\frac {\phi ^{2}}{2!}}\int \phi \operatorname {d} S_{0}+{\frac {\phi }{1}}\int {\frac {\phi ^{2}}{2!}}\operatorname {d} S_{0}-\int {\frac {\phi ^{3}}{3!}}\operatorname {d} S_{0},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \\S_{n}=&{\frac {\phi ^{n}}{n!}}S_{0}-{\frac {\phi ^{n-1}}{(n-1)!}}\int \phi \operatorname {d} S_{0}+\ldots \pm {\frac {\phi }{1!}}\int {\frac {\phi ^{n-1}}{(n-1)!}}\operatorname {d} S_{0}\mp \int {\frac {\phi ^{n}}{2!}}\operatorname {d} S_{0}.\\\end{aligned}}$

La série infinie des arcs de rangs pairs, pris avec leurs signes, est

$S_{0}-S_{2}+S_{4}-S_{6}+\ldots$

Si l’on y substitue pour $S_{0},S_{2},S_{4},\ldots$ les valeurs ci-dessus, il viendra ; en réunissant ce qui multiplie chaque intégrale,

$S_{0}-S_{2}+S_{4}-S_{6}+\ldots =$

$\left(1-{\frac {\phi ^{2}}{2!}}+{\frac {\phi ^{4}}{4!}}-{\frac {\phi ^{6}}{6!}}+\ldots \right)\left(S_{0}-\int {\frac {\phi ^{2}}{2!}}\operatorname {d} S_{0}+\int {\frac {\phi ^{4}}{4!}}\operatorname {d} S_{0}-\ldots \right)$

$+\left({\frac {\phi }{1}}-{\frac {\phi ^{3}}{3!}}+{\frac {\phi ^{5}}{5!}}-\ldots \right)\left(\int {\frac {\phi }{1}}\operatorname {d} S_{0}-\int {\frac {\phi ^{3}}{3!}}\operatorname {d} S_{0}+\int {\frac {\phi ^{5}}{5!}}\operatorname {d} S_{0}-\ldots \right)\,;$

c’est-à-dire,

$S_{0}-S_{2}+S_{4}-S_{6}+\ldots =$

$\left(S_{0}-\int {\frac {\phi ^{2}}{2!}}\operatorname {d} S_{0}+\int {\frac {\phi ^{4}}{4!}}\operatorname {d} S_{0}-\int {\frac {\phi ^{6}}{6!}}\operatorname {d} S_{0}+\ldots \right)\operatorname {Cos} .\phi$

$+\left(\int {\frac {\phi }{1}}\operatorname {d} S_{0}-\int {\frac {\phi ^{3}}{3!}}\operatorname {d} S_{0}+\int {\frac {\phi ^{5}}{5!}}\operatorname {d} S_{0}-\ldots \right)\operatorname {Sin} .\phi \,;$