Page:Lagrange - Œuvres (1867) vol. 1.djvu/685

${\begin{aligned}{\mathfrak {P}}_{3}&=3\left(aa'{\frac {{\mathfrak {Q}}_{1}}{2}}-a'^{2}{\mathfrak {P}}_{1}\right),\\{\mathfrak {Q}}_{3}&=3\left(aa'{\frac {2{\mathfrak {P_{1}+R_{1}}}}{2}}\,-a'^{2}{\mathfrak {Q}}_{1}\right),\\{\mathfrak {R}}_{3}&=3\left(aa'{\frac {\mathfrak {Q_{1}+S_{1}}}{2}}\,\ \ -a'^{2}{\mathfrak {R}}_{1}\right),\\{\mathfrak {S}}_{3}&=3\left(aa'{\frac {\mathfrak {R_{1}+T_{1}}}{2}}\quad -a'^{2}{\mathfrak {S}}_{1}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

75. Cela posé, on aura d’abord

${\begin{aligned}{\frac {r}{v^{3}}}=&a\left({\mathfrak {A_{1}+B_{1}}}\cos \theta +{\mathfrak {C}}_{1}\cos 2\theta +{\mathfrak {D}}_{1}\cos 3\theta +\ldots \right)\\&+iy\ a\left[{\mathfrak {A_{1}+P_{2}}}+({\mathfrak {B_{1}+Q_{2}}})\cos \theta +({\mathfrak {C_{1}+R_{2}}})\cos 2\theta \right.\\&\ \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.+({\mathfrak {D_{1}+S_{2}}})\cos 3\theta +\ldots \right]\\&+iy'a\left({\mathfrak {P_{3}+Q_{3}}}\cos \theta +{\mathfrak {R}}_{3}\cos 2\theta +{\mathfrak {S}}_{3}\cos 3\theta +\ldots \right),\end{aligned}}$

et de même

${\begin{aligned}{\frac {r'}{v^{3}}}=&a'\left({\mathfrak {A_{1}+B_{1}}}\cos \theta +{\mathfrak {C}}_{1}\cos 2\theta +{\mathfrak {D}}_{1}\cos 3\theta +\ldots \right)\\&+iy\ a'\left({\mathfrak {P_{2}+Q_{2}}}\cos \theta +{\mathfrak {R}}_{2}\cos 2\theta +{\mathfrak {S}}_{2}\cos 3\theta +\ldots \right),\\&+iy'a'\left[{\mathfrak {A_{1}+P_{3}}}+({\mathfrak {B_{1}+Q_{3}}})\cos \theta +({\mathfrak {C_{1}+R_{3}}})\cos 2\theta \right.\\&\ \ \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \left.+({\mathfrak {D_{1}+S_{3}}})\cos 3\theta +\ldots \right]\\\end{aligned}}$

Donc, multipliant cette dernière quantité par $\cos \theta ,$ on aura

${\begin{aligned}{\frac {r'}{v^{3}}}\cos \theta =&a'\left[{\frac {{\mathfrak {B}}_{1}}{2}}+{\mathfrak {\left(A_{1}+{\frac {C_{1}}{2}}\right)}}\cos \theta +{\frac {\mathfrak {B_{1}+D_{1}}}{2}}\cos 2\theta +\ldots \right]\\&+iy\ a'\left[{\frac {{\mathfrak {Q}}_{2}}{2}}+{\mathfrak {\left(P_{2}+{\frac {R_{2}}{2}}\right)}}\cos \theta +{\frac {\mathfrak {Q_{2}+S_{2}}}{2}}\cos 2\theta +\ldots \right]\\&+iy'a'\left[{\mathfrak {\frac {B_{1}+Q_{3}}{2}}}+{\mathfrak {\left(A_{1}+P_{3}+{\frac {C_{1}+R_{3}}{2}}\right)}}\cos \theta \right.\\&\ \qquad \qquad \qquad \qquad +\left.{\mathfrak {\frac {B_{1}+Q_{3}+D_{1}++S_{3}}{2}}}\cos 2\theta +\ldots \right].\end{aligned}}$

Or, en négligeant les termes de l’ordre de $i^{2},$

${\frac {r'}{u'^{3}}}={\frac {1}{r'^{2}\left(1+q'^{2}\right)^{\frac {3}{2}}}}={\frac {1-2y'}{a'^{2}}},$