Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 12.djvu/183

$f$ étant égal à ${\sqrt {\left({\frac {\partial \lambda '}{\partial x'}}\right)^{2}+\left({\frac {\partial \lambda '}{\partial y'}}\right)^{2}+\left({\frac {\partial \lambda '}{\partial z'}}\right)^{2}}}.$ L’équation à la surface du sphéroïde est de cette forme

$x'^{2}+y'^{2}+z'^{2}=1+2q,$

$q$ étant une fonction très petite de $x',y',z',$ dont nous négligeons le carré ; on a donc

$\lambda '=x'^{2}+y'^{2}+z'^{2}-1-2q,$

ce qui change les expressions des trois forces précédentes dans celles-ci :

${\begin{aligned}&-{\frac {\alpha gyr^{2}d\mu d\varpi }{f}}\left(x'-{\frac {\partial q}{\partial x'}}\right),\\&-{\frac {\alpha gyr^{2}d\mu d\varpi }{f}}\left(y'-{\frac {\partial q}{\partial y'}}\right),\\&-{\frac {\alpha gyr^{2}d\mu d\varpi }{f}}\left(z'-{\frac {\partial q}{\partial z'}}\right).\end{aligned}}$

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

${\begin{aligned}{\frac {d\mathrm {N} }{dt}}\ \,=&\operatorname {S} {\frac {\alpha gyr^{2}d\mu d\varpi }{f}}\left(y'{\frac {\partial q}{\partial z'}}-z'{\frac {\partial q}{\partial y'}}\right),\\{\frac {d\mathrm {N} '}{dt}}\ =&\operatorname {S} {\frac {\alpha gyr^{2}d\mu d\varpi }{f}}\left(z'{\frac {\partial q}{\partial x'}}-x'{\frac {\partial q}{\partial z'}}\right),\\{\frac {d\mathrm {N} ''}{dt}}=&\operatorname {S} {\frac {\alpha gyr^{2}d\mu d\varpi }{f}}\left(x'{\frac {\partial q}{\partial y'}}-y'{\frac {\partial q}{\partial x'}}\right).\end{aligned}}$

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

${\frac {\partial q}{\partial x'}}dx'+{\frac {\partial q}{\partial y'}}dy'+{\frac {\partial q}{\partial z'}}dz'={\frac {\partial q}{\partial r}}dr+{\frac {\partial q}{\partial \mu }}d\mu +{\frac {\partial q}{\partial \varpi }}d\varpi .$

Or on a

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