Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 4.djvu/218

les formules trigonométriques donnent

${\begin{aligned}\sin \lambda \sin \Gamma =&\sin({\rm {A-\theta )\sin \Pi ,}}\\\sin \lambda \cos \Gamma =&{\rm {\sin \varpi \cos(A-\theta )+\cos \varpi \sin(A-\theta )\cos \Pi ,}}\\\cos \lambda =&{\rm {\cos \varpi \cos(A-\theta )-\sin \varpi \sin(A-\theta )\cos \Pi ,}}\\\sin \gamma \sin(\Psi +\Gamma )=&\sin \theta \sin \Pi ,\\\sin \gamma \cos(\Psi +\Gamma )=&-\sin \varpi \cos \theta +\cos \varpi \sin \theta \cos \Pi ,\\\cos \gamma =&\quad \cos \varpi \cos \theta +\sin \varpi \sin \theta \cos \Pi .\end{aligned}}$

En faisant donc

${\rm {K\sin(A-\theta )\cos(A-\theta )=K'\sin \theta \cos \theta ,}}$

ce qui donne pour déterminer $\theta$ l’équation

$\operatorname {tang} 2\theta ={\rm {{\frac {K\sin 2A}{K'+K\cos 2A}},}}$

on aura

${\begin{aligned}{\frac {d\varpi }{dv}}=&{\rm {-{\frac {1}{2}}\left[K\sin ^{2}(A-\theta )+K'\sin ^{2}\theta \right]\sin \varpi \sin 2\Pi ,}}\\{\frac {d\Pi }{dv}}=&\qquad {\rm {\left[K\cos ^{2}(A-\theta )+K'\cos ^{2}\theta \right]\cos \varpi }}\\&\quad {\rm {-\left[K\sin ^{2}(A-\theta )+K'\sin ^{2}\theta \right]\cos \varpi \cos 2\Pi .}}\end{aligned}}$

Soit donc

${\begin{aligned}q=&{\rm {{\frac {1}{4}}\left(K+K'-{\sqrt {K^{2}+2KK'\cos 2A+K'^{2}}}\right),}}\\p=&{\rm {{\frac {1}{2}}\left(K+K'+{\sqrt {K^{2}+2KK'\cos 2A+K'^{2}}}\right)-q\,;}}\end{aligned}}$

on aura

${\begin{aligned}{\frac {d\varpi }{dv}}=&-q\sin \varpi \sin 2\Pi ,\\{\frac {d\Pi }{dv}}=&p\cos \varpi -q\cos \varpi \cos 2\Pi ,\end{aligned}}$

d’où l’on tire

${\frac {d\varpi \cos \varpi }{\sin \varpi }}=-{\frac {qd\Pi \sin 2\Pi }{p-q\cos 2\Pi }}.$