Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 1.djvu/264

de ces deux produits ; nous aurons ainsi

$0=\operatorname {tang} \theta \left(\cos \alpha {\frac {d^{2}x}{dt^{2}}}+\sin \alpha {\frac {d^{2}y}{dt^{2}}}\right)+\operatorname {tang} \theta {\frac {x\cos \alpha +y\sin \alpha }{r^{3}}}-{\frac {d^{2}z}{dt^{2}}}-{\frac {z}{r^{3}}}.$

Cette équation deviendra, en y substituant pour $x,y,z$ leurs valeurs,

$0=\operatorname {tang} \theta \left[\left({\frac {d^{2}x'}{dt^{2}}}+{\frac {x'}{r^{3}}}\right)\cos \alpha +\left({\frac {d^{2}y'}{dt^{2}}}+{\frac {y'}{r^{3}}}\right)\sin \alpha \right]-{\frac {2\left({\frac {d\theta }{dt}}\right)\left({\frac {d\rho }{dt}}\right)}{\cos ^{2}\theta }}$

$-\rho \left[{\frac {\left({\frac {d^{2}\theta }{dt^{2}}}\right)}{\cos ^{2}\theta }}+{\frac {2\left({\frac {d\theta }{dt}}\right)^{2}\sin \theta }{\cos ^{3}\theta }}+\left({\frac {d\alpha }{dt}}\right)^{2}\operatorname {tang} \theta \right]\,;$

or on a

${\begin{aligned}&\left({\frac {d^{2}x'}{dt^{2}}}+{\frac {x'}{r^{3}}}\right)\cos \alpha +\left({\frac {d^{2}y'}{dt^{2}}}+{\frac {y'}{r^{3}}}\right)\sin \alpha \\\\&\qquad \qquad \qquad \qquad \qquad \qquad =(x'\cos \alpha +y'\sin \alpha )\left({\frac {1}{r^{3}}}-{\frac {1}{\mathrm {R} ^{3}}}\right)\\\\&\qquad \qquad \qquad \qquad \qquad \qquad =\mathrm {R} \cos(\mathrm {A} -\alpha )\left({\frac {1}{r^{3}}}-{\frac {1}{\mathrm {R} ^{3}}}\right)\,;\end{aligned}}$

partant

(2)

$\left\{{\begin{aligned}\left({\frac {d\rho }{dt}}\right)=&-{\frac {1}{2}}\rho \left[{\frac {\frac {d^{2}\theta }{dt^{2}}}{\frac {d\theta }{dt}}}+2\left({\frac {d\theta }{dt}}\right)\operatorname {tang} \theta +{\frac {\left({\frac {d\alpha }{dt}}\right)^{2}\sin \theta \cos \theta }{\left({\frac {d\theta }{dt}}\right)}}\right]\\\\&+{\frac {\mathrm {R} \sin \theta \cos \theta \cos(\mathrm {A} -\alpha )}{2\left({\frac {d\theta }{dt}}\right)}}\left({\frac {1}{r^{3}}}-{\frac {1}{\mathrm {R} ^{3}}}\right)\,;\end{aligned}}\right.$

si l’on retranche cette valeur de $\left({\frac {d\rho }{dt}}\right)$ de la première, et que l’on suppose

$\mu '={\frac {\left({\frac {d\alpha }{dt}}\right)\left({\frac {d^{2}\theta }{dt^{2}}}\right)-\left({\frac {d\theta }{dt}}\right)\left({\frac {d^{2}\alpha }{dt^{2}}}\right)+2\left({\frac {d\alpha }{dt}}\right)\left({\frac {d\theta }{dt}}\right)^{2}\operatorname {tang} \theta +\left({\frac {d\alpha }{dt}}\right)^{3}\sin \theta \cos \theta }{\left({\frac {d\alpha }{dt}}\right)\sin \theta \cos \theta \cos(\mathrm {A} -\alpha )+\left({\frac {d\theta }{dt}}\right)\sin(\mathrm {A} -\alpha )}}$

on aura

(3)

$\rho ={\frac {\mathrm {R} }{\mu '}}\left({\frac {1}{r^{3}}}-{\frac {1}{\mathrm {R} ^{3}}}\right).$