Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 2.djvu/377

${\rm {R}}^{2}d{\rm {R}}d\mu d\varpi$ pour $dm$,

${\frac {d{\rm {N}}}{dt}}=\iiint \alpha {\rm {R}}^{4}d{\rm {Rd\mu }}d\varpi \left(x'{\frac {\partial f}{\partial y'}}-y'{\frac {\partial f}{\partial x'}}\right).$

La profondeur de la mer étant supposée très-petite, et le rayon ${\rm {R}}$ étant à très-peu près égal à l’unité, on a, relativement à la mer,

$\int {\rm {R}}^{4}d{\rm {R}}=\gamma ,$

et par conséquent

${\frac {d{\rm {N}}}{dt}}=\iint \alpha \gamma d\mu d\varpi \left(x'{\frac {\partial f}{\partial y'}}-y'{\frac {\partial f}{\partial x'}}\right).$

On trouvera de la même manière

${\begin{aligned}{\frac {d{\rm {N'}}}{dt}}&=\iint \alpha \gamma d\mu d\varpi \left(x'{\frac {\partial f}{\partial z'}}-z'{\frac {\partial f}{\partial x'}}\right),\\{\frac {d{\rm {N''}}}{dt}}&=\iint \alpha \gamma d\mu d\varpi \left(y'{\frac {\partial f}{\partial z'}}-z'{\frac {\partial f}{\partial y'}}\right).\end{aligned}}$

En transformant, par le n^o 10, les différences partielles en d’autres relatives aux variables ${\rm {R,\mu }}$ et $\varpi ,$ on aura

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

ce qui donne

${\begin{aligned}{\frac {d{\rm {N'\sin \varphi +d{\rm {N''\cos \varphi }}}}}{dt}}&=\iint \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\sin(\varphi +\varpi ){\frac {\partial f}{\partial \mu }},\\&-\iint \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}\cos(\varphi +\varpi ){\frac {\partial f}{\partial \varpi }},\\{\frac {d{\rm {N'\cos \varphi -d{\rm {N''\sin \varphi }}}}}{dt}}&=\iint \alpha \gamma d\mu d\varpi {\sqrt {1-\mu ^{2}}}\cos(\varphi +\varpi ){\frac {\partial f}{\partial \mu }}\\&+\iint \alpha \gamma d\mu d\varpi {\frac {\mu }{\sqrt {1-\mu ^{2}}}}\sin(\varphi +\varpi ){\frac {\partial f}{\partial \varpi }},\end{aligned}}$