Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/92

ce qui donne

${\frac {\partial \theta '}{\partial \theta }}={\frac {\sin \theta \cos ^{2}p+\sin ^{2}p\sin(\theta -2q)}{\sin \theta '}}\,;$

si l’on différence ensuite l’équation

$\sin(\varpi '-\varpi )={\frac {2\sin p\cos p\sin q}{\sin \theta '}}$

par rapport à $\varpi '$ et $\theta ,$ on aura

$\cos(\varpi '-\varpi )d\varpi '=-{\frac {2\sin p\cos p\sin q\cos \theta '}{\sin ^{2}\theta '}}{\frac {\partial \theta '}{\partial \theta }}d\theta \,;$

d’où l’on tire, en substituant au lieu de ${\frac {\partial \theta '}{\partial \theta }}$ sa valeur,

${\frac {\partial \varpi '}{\partial \theta }}=-{\frac {2\sin p\cos p\sin q\cos \theta '}{\sin ^{2}\theta '\cos(\varpi '-\varpi )}}\left[\sin \theta \cos ^{2}p+\sin ^{2}p\sin(\theta -2q)\right],$

${\begin{aligned}&{\frac {\partial \mathrm {A} }{\partial \theta }}=-{\frac {2}{3}}\alpha \pi {\frac {\partial \mu }{\partial \theta }}\\&+\alpha \int _{0}^{\pi }\int _{0}^{\pi }\left\{{\frac {\partial \mu '}{\partial \theta '}}{\frac {\sin \theta \cos ^{2}p+\sin ^{2}p\sin(\theta -2q)}{\sin \theta '}}\sin p\right.\\&\left.-{\frac {\partial \mu '}{\partial \varpi '}}{\frac {2\sin ^{2}p\cos p\sin q\cos \theta '}{\sin ^{3}\theta '\cos(\varpi '-\varpi )}}\left[\sin \theta \cos ^{2}p+\sin ^{2}p\sin(\theta -2q)\right]\right\}dpdq\end{aligned}}$

partant,

${\begin{aligned}{\frac {\partial \mathrm {A} }{\partial \theta }}-{\frac {1}{2}}\mathrm {B} =&-{\frac {2}{3}}\alpha \pi {\frac {\partial \mu }{\partial \theta }}\\&+\alpha \int _{0}^{\pi }\int _{0}^{\pi }\left(\sin ^{2}p-\cos ^{2}p\right)\left\{{\frac {\partial \mu '}{\partial \theta '}}{\frac {\sin p\sin(\theta -2q)}{\sin \theta '}}\right.\\&\qquad \qquad -{\frac {\partial \mu '}{\partial \varpi '}}\left[{\frac {\cos p\cos q}{\sin \theta '\cos(\varpi '-\varpi )}}\right.\\&\qquad \qquad \left.\left.+{\frac {2\sin ^{2}p\cos p\sin q\cos \theta '\sin(\theta -2q)}{\sin ^{3}\theta '\cos(\varpi '-\varpi )}}\right]\right\}dpdq.\end{aligned}}$

Maintenant on a

${\frac {\partial \mu '}{\partial q}}={\frac {\partial \mu '}{\partial \theta '}}{\frac {\partial \theta '}{\partial q}}+{\frac {\partial \mu '}{\partial \varpi '}}{\frac {\partial \varpi '}{\partial q}}\,;$

or, si l’on différentie l’équation

$\cos \theta '=\cos \theta \cos ^{2}p+\sin ^{2}p\cos(\theta -2q),$