Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 3.djvu/48

pour diviseur,

$\delta v={\frac {3(3-i)m'n^{2}}{\left[i(n'-n)+3n\right]^{2}}}$

$\times \left\{{\begin{aligned}&\left[{\rm {P}}'+{\frac {2ad{\rm {P}}}{\left[i(n'-n)+3n\right]dt}}-{\frac {3ad^{2}{\rm {P'}}}{\left[i(n'-n)+3n\right]^{2}dt^{2}}}\right]\\&\qquad \times \sin \left[i(n't-nt+3\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\\-&\left[{\rm {P}}+{\frac {2ad{\rm {P'}}}{\left[i(n'-n)+3n\right]dt}}-{\frac {3ad^{2}{\rm {P}}}{\left[i(n'-n)+3n\right]^{2}dt^{2}}}\right]\\&\qquad \times \cos \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\end{aligned}}\right\}$

$-{\frac {2m'n}{i(n'-n)+3n}}\left[{\begin{aligned}&a^{2}{\frac {\partial P}{\partial a}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\\-&a^{2}{\frac {\partial P'}{\partial a}}\sin \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\end{aligned}}\right]$

$-{\frac {e{\rm {H}}}{2}}\sin \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon -\varpi +{\rm {A}}\right].$

L’équation différentielle

$0={\frac {d^{2}(r\delta r)}{dt^{2}}}+{\frac {\mu .r\delta r}{r^{3}}}+2\int d{\rm {R}}+r{\frac {\partial {\rm {R}}}{\partial r}}$

donne, en ne considérant que les termes qui ont $i(n'-n)+3n$ pour diviseur,

${\frac {r\delta r}{a^{2}}}={\frac {2(i-3)m'n}{i(n'-n)+3n}}\left\{{\begin{aligned}&a{\rm {P}}\sin \left[i(n't-nt+3\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\\+&a{\rm {P'}}\cos \left[i(n't-nt+3\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\end{aligned}}\right\}$

${\begin{aligned}&-{\frac {3}{2}}e{\rm {H}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon -\varpi +{\rm {A}}\right]\\&+{\frac {1}{2}}e{\rm {H}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon +\varpi +{\rm {A}}\right].\end{aligned}}$

En réunissant cette équation à celle-ci

${\frac {r\delta r}{a^{2}}}={\rm {H}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+2nt+2\varepsilon +{\rm {A}}\right],$

on en tirera

${\begin{aligned}{\frac {\delta r}{a}}&={\rm {H}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+2nt+2\varepsilon +{\rm {A}}\right]\\&-e{\rm {H}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+3nt+3\varepsilon -\varpi +{\rm {A}}\right]\\&+e{\rm {H}}\cos \left[i(n't-nt+\varepsilon '-\varepsilon )+nt+\varepsilon +\varpi +{\rm {A}}\right]\\&+{\frac {2(i-3)m'n}{i(n'-n)+3n}}\left\{{\begin{aligned}&a{\rm {P}}\sin \left[i(n't-nt+3\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\\+&a{\rm {P'}}\cos \left[i(n't-nt+3\varepsilon '-\varepsilon )+3nt+3\varepsilon \right]\end{aligned}}\right\}.\end{aligned}}$

Cette valeur de ${\frac {\delta r}{a}}$ introduit dans $\delta v$ une inégalité dépendante de l’angle $i(n't-nt+3\varepsilon '-\varepsilon )+nt+\varepsilon ,$ et qui a pour diviseur $i(n'-n)+3n.$