Page:Lagrange - Œuvres (1867) vol. 1.djvu/687

Maintenant on aura

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

$\left.+\left({\frac {\mathfrak {B_{1}+Q_{3}}}{2}}-{\frac {\mathfrak {D_{1}+S_{3}}}{2}}\right)\sin 2\theta +\ldots \right]$

Donc, si l’on multiplie ces deux quantités par $r=a(i+iy),$ et qu’on fasse

${\begin{aligned}{\mathfrak {A}}_{3}&=a^{2}a'{\frac {2{\mathfrak {A_{1}-C_{1}}}}{2}}-{\frac {a^{2}}{a'^{2}}},\\{\mathfrak {B}}_{3}&=a^{2}a'{\frac {\mathfrak {B_{1}-D_{1}}}{2}},\\{\mathfrak {C}}_{3}&=a^{2}a'{\frac {\mathfrak {C_{1}-E_{1}}}{2}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \,;\\{\mathfrak {P}}_{6}&=a^{2}a'\left({\frac {2{\mathfrak {A_{1}-C_{1}}}}{2}}+{\frac {2{\mathfrak {P_{2}-R_{2}}}}{2}}\right)-{\frac {a^{2}}{a'^{2}}},\\{\mathfrak {Q}}_{6}&=a^{2}a'\left({\frac {\mathfrak {B_{1}-D_{1}}}{2}}+{\frac {\mathfrak {Q_{2}-S_{2}}}{2}}\right),\\{\mathfrak {R}}_{6}&=a^{2}a'\left({\frac {\mathfrak {C_{1}-E_{1}}}{2}}\ \ +{\frac {\mathfrak {R_{2}-T_{2}}}{2}}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\{\mathfrak {P}}_{7}&=a^{2}a'\left({\frac {2{\mathfrak {A_{1}-C_{1}}}}{2}}+{\frac {2{\mathfrak {P_{3}-R_{3}}}}{2}}\right)+{\frac {2a^{2}}{a'^{2}}},\\{\mathfrak {Q}}_{7}&=a^{2}a'\left({\frac {\mathfrak {B_{1}-D_{1}}}{2}}+{\frac {\mathfrak {Q_{3}-S_{3}}}{2}}\right),\\{\mathfrak {R}}_{7}&=a^{2}a'\left({\frac {\mathfrak {C_{1}-E_{1}}}{2}}\ \ \,+{\frac {\mathfrak {R_{3}-T_{3}}}{2}}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

on aura (numéro cité)

$\mathrm {Q} ={\frac {\mathrm {J} '}{a}}\left({\mathfrak {A}}_{3}\sin \theta +{\mathfrak {B}}_{3}\sin 2\theta +{\mathfrak {C}}_{3}\sin 3\theta +\ldots \right)$