Page:Annales de mathématiques pures et appliquées, 1825-1826, Tome 16.djvu/130

En désignant par $m$ un nombre entier positif, plus petit que $n,$ on conclura de là

${\overline {\mathrm {OP_{1}} }}^{2m}+{\overline {\mathrm {OP_{2}} }}^{2m}+{\overline {\mathrm {OP_{3}} }}^{2m}+\ldots +{\overline {\mathrm {OP_{n}} }}^{2m}$

$=\left\{{\begin{aligned}&\left[\left(k^{2}+r^{2}\right)-2kr\operatorname {Cos} .a\right]^{m}\\+&\left[\left(k^{2}+r^{2}\right)-2kr\operatorname {Cos} .(a+x)\right]^{m}\\+&\left[\left(k^{2}+r^{2}\right)-2kr\operatorname {Cos} .(a+2x)\right]^{m}\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\+&\left[\left(k^{2}+r^{2}\right)-2kr\operatorname {Cos} .(a+{\overline {n-1}}.x)\right]^{m}\end{aligned}}\right\}$

$=\left\{{\begin{aligned}&\left(k^{2}+r^{2}\right)^{m}-2{\frac {m}{1}}kr\left(k^{2}+r^{2}\right)^{m-1}\operatorname {Cos} .a\\&\qquad +4{\frac {m}{1}}.{\frac {m-1}{2}}k^{2}r^{2}\left(k^{2}+r^{2}\right)^{m-2}\operatorname {Cos} .^{2}a\\+&\left(k^{2}+r^{2}\right)^{m}-2{\frac {m}{1}}kr\left(k^{2}+r^{2}\right)^{m-1}\operatorname {Cos} .(a+x)\\&\qquad +4{\frac {m}{1}}.{\frac {m-1}{2}}k^{2}r^{2}\left(k^{2}+r^{2}\right)^{m-2}\operatorname {Cos} .^{2}(a+x)\\+&\left(k^{2}+r^{2}\right)^{m}-2{\frac {m}{1}}kr\left(k^{2}+r^{2}\right)^{m-1}\operatorname {Cos} .(a+2x)\\&\qquad +4{\frac {m}{1}}.{\frac {m-1}{2}}k^{2}r^{2}\left(k^{2}+r^{2}\right)^{m-2}\operatorname {Cos} .^{2}(a+2x)\\+&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\+&\left(k^{2}+r^{2}\right)^{m}-2{\frac {m}{1}}kr\left(k^{2}+r^{2}\right)^{m-1}\operatorname {Cos} .(a+{\overline {n-1}}.x)\\&\qquad +4{\frac {m}{1}}.{\frac {m-1}{2}}k^{2}r^{2}\left(k^{2}+r^{2}\right)^{m-2}\operatorname {Cos} .^{2}(a+{\overline {n-1}}.x)\\\end{aligned}}\right\}$