Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/462

Maintenant, si l’on substitue, au lieu de $\delta \varphi ,\delta \mu '(x+\alpha x'),$ dans l’équation (16) de l’article XI, que l’on néglige les quantités de l’ordre $\alpha ^{2},$ et que l’on considère que, par la nature de $x,$ tous les termes sans $\alpha$ disparaissent d’eux-mêmes, on aura, en faisant, pour abréger,

${\begin{aligned}\mathrm {M} &={\frac {3n^{2}}{(n'-n)^{2}-n^{2}}}\left[\sideset {^{1}}{}{\mathrm {F} }+{\frac {1}{i^{2}}}+{\frac {2n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)\right]\\&-{\frac {2n^{2}.\mathrm {I} }{(n'^{2}-n^{2}}}-{\frac {n}{n'}}\left(\mathrm {C+D} -{\frac {3}{2i^{2}}}\right)+{\frac {n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right),\\\mathrm {\sideset {^{1}}{}M} &={\frac {3n^{2}}{(n'-n)^{2}-n^{2}}}\left[\mathrm {\sideset {^{1}}{}F} +{\frac {1}{i^{2}}}+{\frac {2n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)\right]\\&-{\frac {2n^{2}.\mathrm {\sideset {^{1}}{}I} }{((n'-n)^{2}-n^{2}}}-{\frac {n}{n'-2n}}\left(\mathrm {D-C} +{\frac {1}{2i^{2}}}\right)+{\frac {n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right),\\\mathrm {\sideset {^{2}}{}M} &={\frac {3n^{2}}{(2n'-2n)^{2}-n^{2}}}\left({\frac {1}{2}}\mathrm {\sideset {^{2}}{}F} +{\frac {2n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} \right)\\&-{\frac {2n^{2}.\mathrm {\sideset {^{2}}{}I} }{(2n'-n)^{2}-n^{2}}}-{\frac {n}{2n'-n}}\left(\mathrm {\sideset {^{1}}{}D+\sideset {^{1}}{}C} \right)+{\frac {n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} ,\\\mathrm {\sideset {^{3}}{}M} &={\frac {3n^{2}}{(2n'-2n)^{2}-n^{2}}}\left({\frac {1}{2}}\mathrm {\sideset {^{2}}{}F} +{\frac {2n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} \right)\\&-{\frac {2n^{2}.\mathrm {\sideset {^{3}}{}I} }{(2n'-3n)^{2}-n^{2}}}-{\frac {n}{2n'-3n}}\left(\mathrm {\sideset {^{1}}{}D-\sideset {^{1}}{}C} \right)+{\frac {n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} ,\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

${\begin{aligned}\mathrm {N} =&{\frac {2n^{2}.\mathrm {\sideset {^{1}}{}L} }{(2n'-n)^{2}-n^{2}}}+{\frac {n}{2n'-n}}\left(\mathrm {E+C} -{\frac {2}{i^{2}}}\right),\\\mathrm {\sideset {^{1}}{}N} =&\mathrm {C-E} ,\\\mathrm {\sideset {^{2}}{}N} =&{\frac {2n^{2}.\mathrm {\sideset {^{3}}{}L} }{(3n'-2n)^{2}-n^{2}}}+{\frac {n}{3n'-2n}}\left(\mathrm {\sideset {^{1}}{}E+\sideset {^{1}}{}C} \right),\\\mathrm {\sideset {^{3}}{}N} =&{\frac {2n^{2}.\mathrm {\sideset {^{4}}{}L} }{(\ \ n'-2n)^{2}-n^{2}}}+{\frac {n}{\ \ n'-2n}}\left(\mathrm {\sideset {^{1}}{}E-\sideset {^{1}}{}C} \right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$