Page:Lagrange - Œuvres (1867) vol. 1.djvu/686

et par conséquent

${\frac {r'}{u'^{3}}}\cos \theta ={\frac {1}{a'^{2}}}\cos \theta -2iy'{\frac {1}{a'^{2}}}\cos \theta .$

Donc si l’on fait

${\begin{aligned}{\mathfrak {A}}_{2}&=a^{3}{\mathfrak {A}}_{1}-a^{2}a'{\frac {{\mathfrak {B}}_{1}}{2}},\\{\mathfrak {B}}_{2}&=a^{3}{\mathfrak {B}}_{1}-a^{2}a'{\frac {2{\mathfrak {A_{1}+C_{1}}}}{2}}+{\frac {a^{2}}{a'^{2}}},\\{\mathfrak {C}}_{2}&=a^{3}{\mathfrak {C}}_{1}-a^{2}a'{\frac {\mathfrak {B_{1}+D_{1}}}{2}},\\{\mathfrak {D}}_{2}&=a^{3}{\mathfrak {D}}_{1}-a^{2}a'{\frac {\mathfrak {C_{1}+E_{1}}}{2}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\{\mathfrak {P}}_{4}&=a^{3}({\mathfrak {A_{1}+P_{2}}})-a^{2}a'{\frac {{\mathfrak {Q}}_{2}}{2}},\\{\mathfrak {Q}}_{4}&=a^{3}({\mathfrak {B_{1}+Q_{2}}})-a^{2}a'{\frac {2{\mathfrak {P_{2}+R_{2}}}}{2}},\\{\mathfrak {R}}_{4}&=a^{3}({\mathfrak {C_{1}+R_{2}}})-a^{2}a'{\frac {\mathfrak {Q_{2}+S_{2}}}{2}},\\{\mathfrak {S}}_{4}&=a^{3}({\mathfrak {D_{1}+S_{2}}})-a^{2}a'{\frac {\mathfrak {R_{2}+T_{2}}}{2}},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\\{\mathfrak {P}}_{3}&=a^{3}{\mathfrak {P}}_{3}-a^{2}a'\left({\frac {{\mathfrak {B}}_{1}}{2}}+{\frac {{\mathfrak {Q}}_{3}}{2}}\right),\\{\mathfrak {Q}}_{3}&=a^{3}{\mathfrak {Q}}_{3}-a^{2}a'\left({\frac {2{\mathfrak {A_{1}+C_{1}}}}{2}}+{\frac {2{\mathfrak {P_{3}+R_{3}}}}{2}}\right)-{\frac {2a}{a'^{2}}},\\{\mathfrak {R}}_{3}&=a^{3}{\mathfrak {R}}_{3}-a^{2}a'\left({\frac {\mathfrak {B_{1}+D_{1}}}{2}}+{\frac {\mathfrak {Q_{3}+S_{3}}}{2}}\right),\\{\mathfrak {S}}_{3}&=a^{3}{\mathfrak {S}}_{3}-a^{2}a'\left({\frac {\mathfrak {C_{1}+E_{1}}}{2}}+{\frac {\mathfrak {R_{3}+T_{3}}}{2}}\right),\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

on aura (n^o 67

${\begin{aligned}\mathrm {R} =&{\frac {\mathrm {J} '}{a^{2}}}\left({\mathfrak {A_{2}+B_{2}}}\cos \theta +{\mathfrak {C}}_{2}\cos 2\theta +{\mathfrak {D}}_{2}\cos 3\theta +\ldots \right)\\&+i{\frac {\mathrm {J} '}{a^{2}}}y\ \left({\mathfrak {P_{4}+Q_{4}}}\cos \theta +{\mathfrak {R}}_{4}\cos 2\theta +{\mathfrak {S}}_{4}\cos 3\theta +\ldots \right)\\&+i{\frac {\mathrm {J} '}{a^{2}}}y'\left({\mathfrak {P_{5}+Q_{5}}}\cos \theta +{\mathfrak {R}}_{5}\cos 2\theta +{\mathfrak {S}}_{5}\cos 3\theta +\ldots \right),\end{aligned}}$