Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/370

on aura donc

${\begin{aligned}&\iint \alpha gyd\mu d\varpi \left({\sqrt {1-\mu ^{2}}}\sin \varpi {\frac {\partial \gamma }{\partial \mu }}-{\frac {\mu \cos \varpi }{\sqrt {1-\mu ^{2}}}}{\frac {\partial \gamma }{\partial \varpi }}\right)\\\\=-&\iint \alpha g\gamma d\mu d\varpi \left({\sqrt {1-\mu ^{2}}}\sin \varpi {\frac {\partial y}{\partial \mu }}-{\frac {\mu \cos \varpi }{\sqrt {1-\mu ^{2}}}}{\frac {\partial y}{\partial \varpi }}\right).\end{aligned}}$

On trouvera pareillement

${\begin{aligned}&\iint \alpha gyd\mu d\varpi \left({\sqrt {1-\mu ^{2}}}\cos \varpi {\frac {\partial \gamma }{\partial \mu }}+{\frac {\mu \sin \varpi }{\sqrt {1-\mu ^{2}}}}{\frac {\partial \gamma }{\partial \varpi }}\right)\\\\=-&\iint \alpha g\gamma d\mu d\varpi \left({\sqrt {1-\mu ^{2}}}\cos \varpi {\frac {\partial y}{\partial \mu }}+{\frac {\mu \sin \varpi }{\sqrt {1-\mu ^{2}}}}{\frac {\partial y}{\partial \varpi }}\right)\,;\end{aligned}}$

les expressions précédentes de ${\frac {dN}{dt}},{\frac {dN'}{dt}}$ et ${\frac {dN''}{dt}}$ deviendront ainsi

${\begin{aligned}&{\frac {dN}{dt}}=\iint \alpha \gamma d\mu d\varpi \left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial {\rm {U}}}{\partial \varpi }}\right),\\\\&{\frac {dN'}{dt}}=\iint \alpha \gamma d\mu d\varpi \times \\&\qquad \left[{\sqrt {1-\mu ^{2}}}\cos \varpi \left(g{\frac {\partial y}{\partial \mu }}-{\frac {\partial {\rm {U}}}{\partial \mu }}\right)+{\frac {\mu \sin \varpi }{\sqrt {1-\mu ^{2}}}}\left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial {\rm {U}}}{\partial \varpi }}\right)\right]\\&\qquad -\iint \alpha n^{2}yd\mu d\varpi .\mu {\sqrt {1-\mu ^{2}}}\cos \varpi ,\\\\&{\frac {dN''}{dt}}=\iint \alpha \gamma d\mu d\varpi \times \\&\qquad \left[{\sqrt {1-\mu ^{2}}}\sin \varpi \left(g{\frac {\partial y}{\partial \mu }}-{\frac {\partial {\rm {U}}}{\partial \mu }}\right)-{\frac {\mu \cos \varpi }{\sqrt {1-\mu ^{2}}}}\left(g{\frac {\partial y}{\partial \varpi }}-{\frac {\partial {\rm {U}}}{\partial \varpi }}\right)\right]\\&\qquad -\iint \alpha n^{2}yd\mu d\varpi .\mu {\sqrt {1-\mu ^{2}}}\sin \varpi .\end{aligned}}$

11. Déterminons maintenant l’influence de ces quantités sur les mouvements du sphéroïde terrestre autour de son centre de gravité. Pour cela, reprenons les équations (D’) du n^o 1. Si l’on néglige les quantités très-petites ${\rm {\frac {B-A}{C}}}qrdt,\ {\rm {\frac {C-B}{A}}}rpdt$ et ${\rm {\frac {A-C}{B}}}pqdt\,;$ si, de plus, on observe qu’ayant pris pour les axes des $x'$ des $y'$ et des $z'$ les axes principaux, on a $\varphi =0,\ \theta =0,$ on aura

$dp={\frac {d{\rm {N}}}{\rm {C}}},\qquad dq={\frac {d{\rm {N''}}}{\rm {A}}},\qquad dr=-{\frac {d{\rm {N'}}}{\rm {B}}}.$

On voit d’abord que les termes dépendants de très-petits angles, que