Page:Lagrange - Œuvres (1867) vol. 1.djvu/684

si, après avoir pris la somme d’un nombre quelconque de termes de l’une ou de l’autre série, on en regarde le reste comme une progression géométrique, l’erreur sera toujours moindre que la somme de cette progression ; ainsi il sera aisé de juger de la quantité de l’approximation.

74. Supposons donc

$\left(a^{2}-2aa'\cos \theta +a'^{2}\right)^{-{\frac {3}{2}}}={\mathfrak {A_{1}+B_{1}}}\cos \theta +{\mathfrak {C}}_{1}\cos 2\theta +{\mathfrak {D}}_{1}\cos 3\theta +\ldots$

et

$\left(a^{2}-2aa'\cos \theta +a'^{2}\right)^{-{\frac {3}{2}}}={\mathfrak {P_{1}+Q_{1}}}\cos \theta +{\mathfrak {R}}_{1}\cos 2\theta +{\mathfrak {S}}_{1}\cos 3\theta +\ldots$

et nous aurons

${\frac {1}{v^{3}}}={\mathfrak {A_{1}+B_{1}}}\cos \theta +{\mathfrak {C}}_{1}\cos 2\theta +{\mathfrak {D}}_{1}\cos 3\theta +\ldots$

$-3iy\left[a^{2}{\mathfrak {P}}_{1}-aa'{\frac {{\mathfrak {Q}}_{1}}{2}}+\left(a^{2}{\mathfrak {Q}}_{1}-aa'{\mathfrak {P}}_{1}-aa'{\frac {{\mathfrak {R}}_{1}}{2}}\right)\cos \theta \right.$

$\left.+\left(a^{2}{\mathfrak {R}}_{1}-aa'{\frac {{\mathfrak {Q}}_{1}+{\mathfrak {S}}_{1}}{2}}\right)\cos 2\theta +\ldots \right]$

$-3iy'\left[a'^{2}{\mathfrak {P}}_{1}-aa'{\frac {{\mathfrak {Q}}_{1}}{2}}+\left(a'^{2}{\mathfrak {Q}}_{1}-aa'{\mathfrak {P}}_{1}-aa'{\frac {{\mathfrak {R}}_{1}}{2}}\right)\cos \theta \right.$

$\left.+\left(a'^{2}{\mathfrak {R}}_{1}-aa'{\frac {{\mathfrak {Q}}_{1}+{\mathfrak {S}}_{1}}{2}}\right)\cos 2\theta +\ldots \right],$

ou bien

${\begin{aligned}{\frac {1}{v^{3}}}=&{\mathfrak {A_{1}+B_{1}}}\cos \theta +{\mathfrak {C}}_{1}\cos 2\theta +{\mathfrak {D}}_{1}\cos 3\theta +\ldots \\&+iy\ \left({\mathfrak {P_{2}+Q_{2}}}\cos \theta +{\mathfrak {R}}_{2}\cos 2\theta +{\mathfrak {S}}_{2}\cos 3\theta +\ldots \right)\\&+iy'\left({\mathfrak {P_{3}+Q_{3}}}\cos \theta +{\mathfrak {R}}_{3}\cos 2\theta +{\mathfrak {S}}_{3}\cos 3\theta +\ldots \right),\end{aligned}}$

en faisant, pour abréger,

${\begin{aligned}{\mathfrak {P}}_{2}&=3\left(aa'{\frac {{\mathfrak {Q}}_{1}}{2}}-a^{2}{\mathfrak {P}}_{1}\right),\\{\mathfrak {Q}}_{2}&=3\left(aa'{\frac {2{\mathfrak {P_{1}+R_{1}}}}{2}}\,-a^{2}{\mathfrak {Q}}_{1}\right),\\{\mathfrak {R}}_{2}&=3\left(aa'{\frac {\mathfrak {Q_{1}+S_{1}}}{2}}\,\ \ -a^{2}{\mathfrak {R}}_{1}\right),\\{\mathfrak {S}}_{2}&=3\left(aa'{\frac {\mathfrak {R_{1}+T_{1}}}{2}}\quad -a^{2}{\mathfrak {S}}_{1}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$