Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/365

on aura ainsi, en n’ayant égard qu’à ces forces,

${\begin{aligned}{\frac {dN}{dt}}&=\iint {\frac {2\alpha gy{\rm {R^{2}d\mu d\varpi }}}{f}}\left(x'{\frac {\partial q}{\partial y'}}-y'{\frac {\partial q}{\partial x'}}\right),\\{\frac {dN'}{dt}}&=\iint {\frac {2\alpha gy{\rm {R^{2}d\mu d\varpi }}}{f}}\left(x'{\frac {\partial q}{\partial z'}}-z'{\frac {\partial q}{\partial x'}}\right),\\{\frac {dN''}{dt}}&=\iint {\frac {2\alpha gy{\rm {R^{2}d\mu d\varpi }}}{f}}\left(y'{\frac {\partial q}{\partial z'}}-z'{\frac {\partial q}{\partial y'}}\right).\end{aligned}}$

Rapportons les différences partielles ${\frac {\partial q}{\partial x'}},{\frac {\partial q}{\partial y'}}$ et ${\frac {\partial q}{\partial z'}}$ aux variables ${\rm {R,\mu }}$ et $\varpi .$ Pour cela, nous observerons que l’on a

${\rm {R}}={\sqrt {x^{'2}+y^{'2}+z^{'2}}},\qquad \operatorname {tang} \varpi ={\frac {y'}{x'}},\qquad \mu ={\frac {z'}{\rm {R}}},$

d’où il est facile de conclure

${\begin{aligned}{\frac {\partial q}{\partial x'}}&={\sqrt {1-\mu ^{2}}}\cos \varpi {\frac {\partial q}{\partial {\rm {R}}}}-{\frac {\sin \varpi }{{\rm {R}}{\sqrt {1-\mu ^{2}}}}}{\frac {\partial q}{\partial \varpi }}-{\frac {\mu {\sqrt {1-\mu ^{2}}}\cos \varpi }{\rm {R}}}{\frac {\partial q}{\partial \mu }},\\\\{\frac {\partial q}{\partial y'}}&={\sqrt {1-\mu ^{2}}}\sin \varpi {\frac {\partial q}{\partial {\rm {R}}}}+{\frac {\cos \varpi }{{\rm {R}}{\sqrt {1-\mu ^{2}}}}}{\frac {\partial q}{\partial \varpi }}-{\frac {\mu {\sqrt {1-\mu ^{2}}}\sin \varpi }{\rm {R}}}{\frac {\partial q}{\partial \mu }},\\\\{\frac {\partial q}{\partial z'}}&=\mu {\frac {\partial q}{\partial {\rm {R}}}}+{\frac {1-\mu ^{2}}{\rm {R}}}{\frac {\partial q}{\partial \mu }},\end{aligned}}$

on aura ainsi, en observant que, dans les valeurs de ${\frac {dN}{dt}},{\frac {dN'}{dt}},{\frac {dN''}{dt}},$ on peut, en négligeant le carré de $q,$ supposer ${\rm {R=1}}$ et $f=2,$

${\begin{aligned}{\frac {dN}{dt}}&=\iint \alpha gyd\mu d\varpi {\frac {\partial q}{\partial \varpi }},\\\\{\frac {dN'}{dt}}&=\iint \alpha gyd\mu d\varpi \left({\sqrt {1-\mu ^{2}}}\cos \varpi {\frac {\partial q}{\partial \mu }}+{\frac {\mu \sin \varpi }{\sqrt {1-\mu ^{2}}}}{\frac {\partial q}{\partial \varpi }}\right),\\\\{\frac {dN''}{dt}}&=\iint \alpha gyd\mu d\varpi \left({\sqrt {1-\mu ^{2}}}\sin \varpi {\frac {\partial q}{\partial \mu }}-{\frac {\mu \cos \varpi }{\sqrt {1-\mu ^{2}}}}{\frac {\partial q}{\partial \varpi }}\right).\end{aligned}}$

Déterminons présentement les valeurs de $frac{dN}{dt},{\frac {dN'}{dt}}$ et ${\frac {dN''}{dt}}$ relatives a l’attraction de la couche aqueuse sur le sphéroïde terrestre. Il est clair