Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/17

$d^{2}\mathrm {Z\delta Z} \,;$ et, après avoir effacé ce qui se détruit, et mis pour

$\mathrm {\sin Q'\cos Q',\quad \sin ^{2}Q',\quad \cos ^{2}Q'}$

leurs valeurs

$\mathrm {{\frac {1}{2}}\sin 2Q',\quad {\frac {1}{2}}-{\frac {1}{2}}\cos 2Q',\quad {\frac {1}{2}}+{\frac {1}{2}}\cos 2Q'} ,$

on aura

$d^{2}\mathrm {X\delta X} +d^{2}\mathrm {Y\delta Y} +d^{2}\mathrm {Z\delta Z}$

$=\left\{{\begin{aligned}&r^{2}\cos ^{2}\mathrm {P} \left[d(d\omega +\cos \pi d\varepsilon )+{\tfrac {1}{2}}\sin 2\mathrm {Q} '\left(\sin ^{2}\pi d\varepsilon ^{2}-d\pi ^{2}\right)\right.\\&\qquad \qquad \qquad \qquad \qquad \qquad {\Bigl .}+\cos 2\mathrm {Q} '\sin \pi d\pi d\varepsilon {\Bigr ]}\\+&r^{2}\sin \mathrm {P} \cos \mathrm {P} \left[\sin \mathrm {Q} '\left(\sin \pi d^{2}\varepsilon +2\cos \pi d\pi d\varepsilon \right)\right.\\&\qquad \qquad \qquad \qquad \qquad \left.+\cos \mathrm {Q} '\left(d^{2}\pi -\sin \pi \cos \pi d\varepsilon ^{2}\right)\right]\end{aligned}}\right\}\times \delta \omega$

$+\left\{{\begin{aligned}&r^{2}\cos ^{2}\mathrm {P} \left[d\left(\cos \pi d\omega +d\varepsilon -{\tfrac {1}{2}}\sin ^{2}\pi d\varepsilon \right)\right.\\&\quad \qquad \qquad \qquad -\sin 2\mathrm {Q} '\left[\sin ^{2}\pi d\omega d\varepsilon +{\tfrac {1}{2}}d(\sin \pi d\pi )\right]\\&\quad \qquad \qquad \qquad {\Bigl .}+\cos 2\mathrm {Q} '\left[{\tfrac {1}{2}}d\left(\sin ^{2}\pi d\varepsilon \right)-\sin \pi d\omega d\pi \right]{\Bigr ]}\\+&r^{2}\sin \mathrm {P} \cos \mathrm {P} {\Bigl [}\sin \mathrm {Q} '\left[\sin \pi d^{2}\omega +d(\sin 2\pi d\varepsilon )\right]{\Bigr .}\\&\qquad \quad \ \ {\Bigl .}+\cos \mathrm {Q} '\left[\sin \pi d\omega ^{2}+\sin 2\pi d\varepsilon d\omega +d(\cos \pi d\pi )\right]{\Bigr ]}\\+&r^{2}\sin ^{2}\mathrm {P} \left[d\left(\sin ^{2}\pi d\varepsilon \right)\right]\end{aligned}}\right\}\times \delta \varepsilon$

$+\left\{{\begin{aligned}&r^{2}\cos ^{2}\mathrm {P} \left[\sin \pi d\omega d\varepsilon +{\tfrac {1}{2}}\sin \pi \cos \pi d\varepsilon ^{2}\right.\\&\ \ \ \qquad {\Bigl .}+\sin 2\mathrm {Q} '\left(\sin \pi d\omega d\varepsilon +{\tfrac {1}{2}}\sin \pi \cos \pi d\varepsilon ^{2}+{\tfrac {1}{2}}d^{2}\pi \right){\Bigr ]}\\&\ \ \ \qquad {\Bigl .}-\cos 2\mathrm {Q} '\left(\sin \pi d\omega d\varepsilon +{\tfrac {1}{2}}\sin \pi \cos \pi d\varepsilon ^{2}+{\tfrac {1}{2}}d^{2}\pi \right){\Bigr ]}\\+&r^{2}\sin \mathrm {P} \cos \mathrm {P} {\Bigl [}-\sin \mathrm {Q} '\left(2\cos \pi d\omega d\varepsilon +\cos 2\pi d\varepsilon ^{2}+{\tfrac {1}{2}}d^{2}\pi \right){\Bigr .}\\&\qquad \qquad \qquad \qquad {\Bigl .}+\cos \mathrm {Q} '\left(d^{2}\omega +\cos \pi d^{2}\varepsilon \right){\Bigr ]}\\+&r^{2}\sin ^{2}\mathrm {P} \left[d^{2}\pi -\sin \pi \cos \pi d\varepsilon ^{2}\right]\end{aligned}}\right\}\times \delta \pi .$

On multipliera cette quantité par $\alpha ,$ et l’on en prendra l’intégrale en faisant varier seulement $r,\mathrm {P,Q}$ (Article v) ; on aura ainsi la valeur de

$\int \alpha \left(d^{2}\mathrm {X\delta X} +d^{2}\mathrm {Y\delta Y} +d^{2}\mathrm {Z\delta Z} \right),$

qu’il faudra substituer dans l’équation (A), Article III.