Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/296

donc

$r'^{2}={\frac {\mathrm {R} ^{2}}{i^{2}}},\quad r''^{2}={\frac {\mathrm {R} ^{2}}{i^{2}}}-{\frac {2\mathrm {R} z}{i}}+r^{2}.$

De là on aura

${\begin{aligned}{\frac {1}{r'}}\ \,=&{\frac {i}{\mathrm {R} }},\\{\frac {1}{r''}}\,=&{\frac {i}{\mathrm {R} }}+{\frac {i^{2}\left(2\mathrm {R} z-ir^{2}\right)}{2\mathrm {R} ^{3}}}+{\frac {3i^{3}\left(2\mathrm {R} z-ir^{2}\right)^{2}}{8\mathrm {R} ^{5}}}+{\frac {5i^{4}\left(2\mathrm {R} z-ir^{2}\right)^{3}}{16\mathrm {R} ^{7}}}+\ldots \\=&{\frac {i}{\mathrm {R} }}+{\frac {i^{2}z}{\mathrm {R} ^{2}}}+{\frac {i^{3}\left(3z^{2}-r^{2}\right)}{2\mathrm {R} ^{3}}}+{\frac {i^{4}\left(5z^{3}-3zr^{2}\right)}{2\mathrm {R} ^{4}}}+\ldots ,\\\\{\frac {1}{r'^{3}}}=&{\frac {i^{3}}{\mathrm {R} ^{3}}},\\{\frac {1}{r''^{3}}}=&{\frac {i^{3}}{\mathrm {R} ^{3}}}+{\frac {3i^{4}\left(2\mathrm {R} z-ir^{2}\right)}{2\mathrm {R} ^{5}}}+{\frac {15i^{5}\left(2\mathrm {R} z-ir^{2}\right)^{2}}{8\mathrm {R} ^{7}}}+{\frac {35i^{6}\left(2\mathrm {R} z-ir^{2}\right)^{3}}{16\mathrm {R} ^{9}}}+\ldots \\=&{\frac {i^{3}}{\mathrm {R} ^{3}}}+{\frac {3i^{4}z}{\mathrm {R} ^{4}}}+{\frac {i^{5}\left(15z^{2}-3r^{2}\right)}{2\mathrm {R} ^{5}}}+{\frac {i^{6}\left(35z^{3}-15zr^{2}\right)}{2\mathrm {R} ^{6}}}+\ldots .\end{aligned}}$

Donc (Article XXII)

${\begin{aligned}p\ \,=&{\frac {\mathrm {R} ^{2}}{i^{2}}}-{\frac {\mathrm {R} z}{i}},\quad p'=-{\frac {\mathrm {R} z}{i}}+r^{2},\quad p''={\frac {\mathrm {R} z}{i}},\\q\ \,=&-{\frac {3i^{4}z}{\mathrm {R} ^{4}}}-{\frac {i^{5}\left(15z^{2}-3r^{2}\right)}{2\mathrm {R} ^{5}}}-{\frac {i^{6}\left(35z^{3}-15zr^{2}\right)}{2\mathrm {R} ^{6}}}-\ldots ,\\q'\,=&{\frac {1}{r^{3}}}-{\frac {i^{3}}{\mathrm {R} ^{3}}}-{\frac {3i^{4}z}{\mathrm {R} ^{4}}}-{\frac {i^{5}\left(15z^{2}-3r^{2}\right)}{2\mathrm {R} ^{5}}}-{\frac {i^{6}\left(35z^{3}-15zr^{2}\right)}{2\mathrm {R} ^{6}}}-\ldots ,\\q''=&-{\frac {1}{r^{3}}}+{\frac {i^{3}}{\mathrm {R} ^{3}}}\,;\end{aligned}}$

et de là

${\begin{aligned}pq\ \ \ =&-{\frac {3i^{2}z}{\mathrm {R} ^{2}}}-{\frac {i^{3}\left(9z^{2}-3r^{2}\right)}{2\mathrm {R} ^{3}}}-{\frac {i^{4}\left(20z^{3}-12zr^{2}\right)}{2\mathrm {R} ^{4}}}-\ldots ,\\p'\,q'\,=&-{\frac {\mathrm {R} z}{ir^{3}}}{\frac {1}{r}}+{\frac {i^{2}z}{\mathrm {R} ^{2}}}+{\frac {i^{3}\left(3z^{2}-r^{2}\right)}{\mathrm {R} ^{3}}}+{\frac {i^{4}\left(15z^{3}-9r^{2}z\right)}{2\mathrm {R} ^{4}}}\\&\qquad \qquad \qquad \qquad \qquad +{\frac {i^{5}\left(35z^{4}-30z^{2}r^{2}+3r^{4}\right)}{2\mathrm {R} ^{5}}}+\ldots ,\\p''q''=&-{\frac {\mathrm {R} z}{ir^{3}}}+{\frac {i^{2}z}{\mathrm {R} ^{2}}}.\end{aligned}}$