Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/130

on aura

${\begin{aligned}y'=&\varepsilon \left(3\cos ^{2}\theta '-1\right)\\&+\left[a\sin(\varphi -\varpi ')+b\cos(\varphi -\varpi ')\right]\\&\qquad \times \sin \theta '\cos \theta '\left(f+f^{(1)}\sin ^{2}\theta '+\ldots +f^{(r)}\sin ^{2r}\theta '\right)\\&+\left[a'\sin(2\varphi -2\varpi ')+b'\cos(2\varphi -2\varpi ')\right]\\&\qquad \times \sin ^{2}\theta '\left(p+p^{(1)}\sin ^{2}\theta '+\ldots +p^{(r)}\sin ^{2r}\theta '\right)\,;\end{aligned}}$

maintenant on a

${\begin{aligned}\sin &(\ \ \varphi -\ \ \varpi ')=\\&\qquad \sin(\ \ \varphi -\ \ \varpi )\cos(\ \ \varpi '-\ \ \varpi )-\cos(\ \ \varphi -\ \ \varpi )\sin(\ \ \varpi '-\ \ \varpi ),\\\cos &(\ \ \varphi -\ \ \varpi ')=\\&\qquad \cos(\ \ \varphi -\ \ \varpi )\cos(\ \ \varpi '-\ \ \varpi )+\sin(\ \ \varphi -\ \ \varpi )\sin(\ \ \varpi '-\ \ \varpi ),\\\sin &(2\varphi -2\varpi ')=\\&\qquad \sin(2\varphi -2\varpi )\cos(2\varpi '-2\varpi )-\cos(2\varphi -2\varpi )\sin(2\varpi '-2\varpi ),\\\cos &(2\varphi -2\varpi ')=\\&\qquad \cos(2\varphi -2\varpi )\cos(2\varpi '-2\varpi )+\sin(2\varphi -2\varpi )\sin(2\varpi '-2\varpi )\,;\end{aligned}}$

donc

${\begin{aligned}y'=&\varepsilon \left(3\cos ^{2}\theta '-1\right)\\&+\left\{\left[a\sin(\varphi -\varpi )+b\cos(\varphi -\varpi )\right]\cos(\varpi '-\varpi )\right.\\&\qquad \qquad \left.+\left[b\sin(\varphi -\varpi )-a\cos(\varphi -\varpi )\right]\sin(\varpi '-\varpi )\right\}\\&\qquad \qquad \qquad \times \sin \theta '\cos \theta '\left(f+f^{(1)}\sin ^{2}\theta '+\ldots +f^{(r)}\sin ^{2r}\theta '\right)\\&+\left\{\left[a'\sin(2\varphi -2\varpi )+b'\cos(2\varphi -2\varpi )\right]\cos(2\varpi '-2\varpi )\right.\\&\qquad \qquad \left.+\left[b'\sin(2\varphi -2\varpi )-a'\cos(2\varphi -2\varpi )\right]\sin(2\varpi '-2\varpi )\right\}\\&\qquad \qquad \qquad \times \sin ^{2}\theta '\left(p+p^{(1)}\sin ^{2}\theta '+\ldots +p^{(r)}\sin ^{2r}\theta '\right).\end{aligned}}$

On tirera facilement, des valeurs de $\cos \theta '$ et de $\sin(\varpi '-\varpi ),$

${\begin{aligned}\cos(\ \ \varpi '-\ \ \varpi )=&{\frac {\sin \theta -2\sin ^{2}p\sin q\cos(\theta -q)}{\sin \theta '}},\\\sin(2\varpi '-2\varpi )=&{\frac {4\sin \theta \sin p\cos ^{3}p\sin q+4\sin ^{3}p\cos p\sin q\sin(\theta -2q)}{\sin ^{2}\theta '}},\\\cos(2\varpi '-2\varpi )=&{\frac {\left[\sin \theta \cos ^{2}p+\sin ^{2}p\sin(\theta -2q)\right]^{2}-4\sin ^{2}p\cos ^{2}p\sin ^{2}q}{\sin ^{2}\theta '}}\,;\end{aligned}}$

on aura, cela posé, en négligeant les termes qu’il est permis de négliger d’après la remarque que nous avons faite ci-dessus,

$\mathrm {C} =\int _{0}^{\pi }\int _{0}^{\pi }{\frac {\cos p}{\sin q}}y'dpdq$

${\begin{aligned}=&\left[b\sin(\varphi -\varpi )-a\cos(\varphi -\varpi )\right]\times \\&\int _{0}^{\pi }\int _{0}^{\pi }2\sin p\cos ^{2}p\cos \theta '\left(f+f^{(1)}\sin ^{2}\theta '+\ldots +f^{(r)}\sin ^{2r}\theta '\right)dpdq\\+&\left[b'\sin(2\varphi -2\varpi )-a'\cos(2\varphi -2\varpi )\right]\times \\&\int _{0}^{\pi }\int _{0}^{\pi }4\sin p\cos ^{4}p\sin \theta \left(p+p^{(1)}\sin ^{2}\theta '+\ldots +p^{(r)}\sin ^{2r}\theta '\right)dpdq\\+&\left[b'\sin(2\varphi -2\varpi )-a'\cos(2\varphi -2\varpi )\right]\times \\&\int _{0}^{\pi }\int _{0}^{\pi }4\sin ^{3}p\cos ^{2}p\sin(\theta -2q)\left(p+p^{(1)}\sin ^{2}\theta '+\ldots +p^{(r)}\sin ^{2r}\theta '\right)dpdq.\end{aligned}}$