Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/222

rotation du corps étant supposé avoir lieu dans le sens $\mathrm {ACB} .$ On aura ainsi, en exécutant les différentiations indiquées dans les équations (L),

$(\mathrm {L} ')\left\{{\begin{aligned}0&={\frac {\psi \mathrm {Y} -\psi '\mathrm {Z} '}{\mathrm {M} a^{2}}}dt^{2}\\&+d^{2}\varepsilon \sin \theta \cos \theta \left(1-{\frac {b^{2}+c^{2}}{2a^{2}}}+{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\\&-d\varepsilon d\theta \left[1-\cos ^{2}\theta \left(1-{\frac {b^{2}+c^{2}}{2a^{2}}}+{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\right]\\&-2d\varepsilon d\varpi \sin \theta \cos \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi +d\varepsilon ^{2}\cos \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi \\&-d^{2}\sin \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi -2d\varpi d\theta \sin \theta \left({\frac {b^{2}+c^{2}}{2a^{2}}}+{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\\&-d\theta ^{2}\cos \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi +d^{2}\varpi \cos \theta {\frac {b^{2}+c^{2}}{2a^{2}}},\\\\0&={\frac {\psi \mathrm {X} i-\psi ''\mathrm {Z} ''}{\mathrm {M} a^{2}}}dt^{2}\\&+d^{2}\varepsilon \cos \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi -2d\varepsilon d\varpi \cos \theta \left({\frac {b^{2}+c^{2}}{2a^{2}}}-{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\\&-d^{2}\varepsilon \sin \theta \cos \theta \left(1-{\frac {b^{2}+c^{2}}{2a^{2}}}+{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\\&-d^{2}\theta \left(1+{\frac {b^{2}+c^{2}}{2a^{2}}}-{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)-2d\varpi d\theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi ,\\\\0&={\frac {\psi '\mathrm {X} '-\psi ''\mathrm {Y} ''}{\mathrm {M} a^{2}}}dt^{2}\\&-d^{2}\varepsilon \left[1+{\frac {b^{2}+c^{2}}{2a^{2}}}-\left(1-{\frac {b^{2}+c^{2}}{2a^{2}}}\right)\sin ^{2}\theta +{\frac {b^{2}-c^{2}}{2a^{2}}}\cos ^{2}\theta \cos 2\varpi \right]\\&+2d\varepsilon d\theta \sin \theta \cos \theta \left(1-{\frac {b^{2}+c^{2}}{2a^{2}}}+{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\\&+2d\varepsilon d\varpi \cos ^{2}\theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi +d^{2}\theta \cos \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi \\&+2d\varpi d\theta \cos \theta \left({\frac {b^{2}+c^{2}}{2a^{2}}}+{\frac {b^{2}-c^{2}}{2a^{2}}}\cos 2\varpi \right)\\&-d^{2}\theta \sin \theta {\frac {b^{2}-c^{2}}{2a^{2}}}\sin 2\varpi +2d^{2}\varpi \sin \theta {\frac {b^{2}+c^{2}}{2a^{2}}}.\end{aligned}}\right.$