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Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/455
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On aura
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{\displaystyle \int {\frac {arr'dt\sin(\varphi '-\varphi )}{\sideset {^{1}}{^{3}}v}}={\frac {1}{n-n'}}\left\{{\begin{aligned}\mathrm {C} &\cos \ \ (n't-nt+\mathrm {B} )\\+{\frac {1}{4}}\sideset {^{1}}{}{\mathrm {C} }&\cos 2(n't-nt+\mathrm {B} )\\+{\frac {1}{9}}\sideset {^{2}}{}{\mathrm {C} }&\cos 3(n't-nt+\mathrm {B} )+\ldots \end{aligned}}\right\}}
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{\displaystyle -\alpha e\left\{{\begin{alignedat}{2}{\frac {\mathrm {C+D} }{n'}}&\cos(\ \ n't&&+\mathrm {B} +\theta )\\+{\frac {\mathrm {D-C} }{n'-2n}}&\cos(\ \ n't&-2nt&+\mathrm {B} -\theta )\\+{\frac {\mathrm {\sideset {^{1}}{}{\mathrm {D} }+\sideset {^{1}}{}{\mathrm {C} }} }{2n'-n}}&\cos(2n't&-\ \ nt&+2\mathrm {B} +\theta )\\+{\frac {\mathrm {\sideset {^{1}}{}{\mathrm {D} }-\sideset {^{1}}{}{\mathrm {C} }} }{2n'-3n}}&\cos(2n't&-3nt&+2\mathrm {B} -\theta )\\+{\frac {\mathrm {\sideset {^{2}}{}{\mathrm {D} }+\sideset {^{2}}{}{\mathrm {C} }} }{3n'-2n}}&\cos(3n't&-2nt&+3\mathrm {B} +\theta )\\+{\frac {\mathrm {\sideset {^{2}}{}{\mathrm {D} }-\sideset {^{2}}{}{\mathrm {C} }} }{3n'-4n}}&\cos(3n't&-4nt&+3\mathrm {B} +\theta )\\+\ldots \ldots &\ldots \ldots &\ldots \ldots &\ldots \ldots \end{alignedat}}\right\}}
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{\displaystyle -\alpha e'\left\{{\begin{alignedat}{2}{\frac {\mathrm {E+C} }{2n'-n}}&\cos(2n't&-nt&+\mathrm {B} +\theta ')\\+{\frac {\mathrm {C-E} }{n}}&\cos(\quad nt&&-\mathrm {B} +\theta '')\\+{\frac {\mathrm {\sideset {^{1}}{}{\mathrm {E} }+\sideset {^{1}}{}{\mathrm {C} }} }{3n'-2n}}&\cos(3n't&-2nt&+2\mathrm {B} +\theta ')\\+{\frac {\mathrm {\sideset {^{1}}{}{\mathrm {E} }-\sideset {^{1}}{}{\mathrm {C} }} }{n'-2n}}&\cos(\ \ n't&-2nt&+2\mathrm {B} -\theta ')\\+{\frac {\mathrm {\sideset {^{2}}{}{\mathrm {E} }+\sideset {^{2}}{}{\mathrm {C} }} }{4n'-3n}}&\cos(4n't&-3nt&+3\mathrm {B} -\theta ')\\+{\frac {\mathrm {\sideset {^{2}}{}{\mathrm {E} }-\sideset {^{2}}{}{\mathrm {C} }} }{2n'-3n}}&\cos(2n't&-3nt&+3\mathrm {B} -\theta ')\\+\ldots \ldots &\ldots \ldots &\ldots \ldots &\ldots \ldots \end{alignedat}}\right\}}
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{\displaystyle +\alpha ^{2}ee'\mathrm {K} t\sin \mathrm {V} +\alpha ^{2}\gamma \gamma '\mathrm {L} t\sin \mathrm {U} .}
Soit encore
(
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;
{\displaystyle {\begin{aligned}(b)-{\frac {i(b_{1})}{2}}\qquad \qquad \qquad =&\ \mathrm {F} ,\\(b_{1})-\ \ i\left[\quad (b\ )+{\frac {1}{2}}(b_{2})\right]=&\sideset {^{1}}{}{\mathrm {F} },\\2(b_{2})-2i\left[{\frac {1}{2}}(b_{1})+{\frac {1}{2}}(b_{3})\right]=&\sideset {^{2}}{}{\mathrm {F} },\\3(b_{3})-3i\left[{\frac {1}{2}}(b_{2})+{\frac {1}{2}}(b_{4})\right]=&\sideset {^{3}}{}{\mathrm {F} },\\\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots &\ldots \,;\end{aligned}}}