Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/23

port au plan de son écliptique ; et cherchons la position du plan de l’équateur de manière que l’on ait

$\int \alpha r^{2}\mathrm {\sin P\cos P\cos Q'} =0,\quad \int \alpha r^{2}\mathrm {\sin P\cos P\sin Q'} =0\,;$

on aura d’abord, en combinant les formules (C) de l’Article VI

${\begin{aligned}\sin \mathrm {P} =&\sin \pi \cos p\sin q'+\cos \pi \sin p,\\\mathrm {\cos P\sin Q'} =&\cos \pi \cos p\sin q'-\sin \pi \sin p,\\\mathrm {\cos P\cos Q'} =&\cos p\cos q'\,;\end{aligned}}$

donc

${\begin{aligned}\mathrm {\sin P\cos P\sin Q'} =&\sin \pi \cos \pi \left(\cos ^{2}p\sin ^{2}q'-\sin ^{2}p\right)\\&\qquad \qquad +\left(\cos ^{2}\pi -\sin ^{2}\pi \right)\cos p\sin p\sin q',\\\mathrm {\sin P\cos P\cos Q'} =&\sin \pi \cos ^{2}p\sin q'\cos q'+\cos \pi \sin p\cos p\cos q'.\end{aligned}}$

et, en mettant pour $q'$ sa valeur $q-\varepsilon$ afin que l’angle $q'$ ait une origine fixe,

${\begin{aligned}\sin &\mathrm {P\cos P\sin Q'} \\=&\sin \pi \cos \pi \left(\cos ^{2}p\sin ^{2}q\cos ^{2}\varepsilon -2\cos ^{2}p\sin q\cos q\sin \varepsilon \cos \varepsilon \right.\\&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.+\cos ^{2}p\cos ^{2}q\sin ^{2}\varepsilon -\sin ^{2}p\right)\\&+\left(\cos ^{2}\pi -\sin ^{2}\pi \right)(\cos p\sin p\sin q\cos \varepsilon -\cos p\sin p\cos q\sin \varepsilon ),\\\\\sin &\mathrm {P\cos P\cos Q'} \\=&\sin \pi \cos ^{2}p\sin q\cos q\left(\cos ^{2}\varepsilon -\sin ^{2}\varepsilon \right)\\&\qquad \qquad \qquad \qquad +\sin \pi \sin \varepsilon \cos \varepsilon \left(\cos ^{2}p\sin ^{2}q-\cos ^{2}p\cos ^{2}q\right)\\&+\cos \pi (\sin p\cos p\sin q\sin \varepsilon +\sin p\cos p\cos q\cos \varepsilon ).\end{aligned}}$

Donc si l’on fait, pour abréger,

${\begin{array}{ll}\int \alpha r^{2}\cos ^{2}p\sin ^{2}q=\mathrm {A} ,&\int \alpha r^{2}\cos ^{2}p\cos ^{2}q=\mathrm {B} ,\\\int \alpha r^{2}\cos ^{2}p\sin q\cos q=\mathrm {C} ,&\int \alpha r^{2}\cos p\sin p\sin q=\mathrm {D} ,\\\int \alpha r^{2}\cos p\sin p\cos q=\mathrm {E} ,&\int \alpha r^{2}\sin ^{2}p=\mathrm {F} ,\\\end{array}}$