Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/460

observera que, par la nature de $y,$ les termes sans $\alpha$ doivent se détruire réciproquement ; de cette manière, en faisant, pour abréger.

${\begin{aligned}\mathrm {I} =&-{\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}{n'}}\left(\mathrm {C+D} -{\frac {3}{2i^{2}}}\right)-{\frac {3n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)+\mathrm {\sideset {^{1}}{}G+\sideset {^{1}}{}F} +{\frac {1}{i^{2}}},\\\mathrm {\sideset {^{1}}{}I} =&-{\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}{n'-2n}}\left(\mathrm {D-C} -{\frac {1}{2i^{2}}}\right)-{\frac {3n}{n'-n}}\left(\mathrm {C} -{\frac {1}{i^{2}}}\right)+\mathrm {\sideset {^{1}}{}G-\sideset {^{1}}{}F} -{\frac {1}{i^{2}}},\\\mathrm {\sideset {^{2}}{}I} =&-{\frac {3n^{2}}{(2n'-2n)^{2}-n^{2}}}\left({\frac {1}{2}}\sideset {^{2}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} \right)\\&+{\frac {2n}{2n'-n}}\left(\mathrm {\sideset {^{1}}{}D+\sideset {^{1}}{}C} \right)-{\frac {3n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C+\sideset {^{2}}{}G+\sideset {^{2}}{}F} ,\\\mathrm {\sideset {^{3}}{}I} =&-{\frac {3n^{2}}{(2n'-2n)^{2}-n^{2}}}\left({\frac {1}{2}}\sideset {^{2}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C} \right)\\&+{\frac {2n}{2n'-n}}\left(\mathrm {\sideset {^{1}}{}D+\sideset {^{1}}{}C} \right)-{\frac {3n}{n'-n}}{\frac {1}{4}}\mathrm {\sideset {^{1}}{}C+\sideset {^{2}}{}G-\sideset {^{2}}{}F} ,\\\mathrm {\sideset {^{4}}{}I} =&-{\frac {3n^{2}}{(3n'-3n)^{2}-n^{2}}}\left({\frac {1}{3}}\sideset {^{3}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{9}}\mathrm {\sideset {^{2}}{}C} \right)\\&+{\frac {2n}{3n'-2n}}\left(\mathrm {\sideset {^{2}}{}D+\sideset {^{2}}{}C} \right)-{\frac {3n}{n'-n}}{\frac {1}{9}}\mathrm {\sideset {^{2}}{}C+\sideset {^{3}}{}G+\sideset {^{3}}{}F} ,\\\mathrm {\sideset {^{5}}{}I} =&-{\frac {3n^{2}}{(3n'-3n)^{2}-n^{2}}}\left({\frac {1}{3}}\sideset {^{3}}{}{\mathrm {F} }+{\frac {2n}{n'-n}}{\frac {1}{9}}\mathrm {\sideset {^{2}}{}C} \right)\\&+{\frac {2n}{3n'-4n}}\left(\mathrm {\sideset {^{2}}{}D-\sideset {^{2}}{}C} \right)-{\frac {3n}{n'-n}}{\frac {1}{9}}\mathrm {\sideset {^{2}}{}C+\sideset {^{3}}{}G-\sideset {^{3}}{}F} ,\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

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