Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/21

${\begin{aligned}&+{\frac {1}{2.3}}\left(1.2.\sigma +2.3.\sigma _{1}{\frac {a}{b}}+3.4.\sigma _{2}{\frac {a^{2}}{b^{2}}}+\ldots \right)\\&+{\frac {1}{2.3.4}}\left(1.2.3.\tau +2.3.4.\tau _{1}{\frac {a}{b}}+3.4.5.\tau _{2}{\frac {a^{2}}{b^{2}}}+\ldots \right)\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\\p\ \,=&{\frac {a}{b}}+\varpi {\frac {a}{b}}+\varpi _{1}{\frac {a^{2}}{b^{2}}}+\varpi _{2}{\frac {a^{3}}{b^{3}}}+\ldots \\&+{\frac {1}{2}}\left(2\rho {\frac {a}{b}}+3\rho _{1}{\frac {a^{2}}{b^{2}}}+4\rho _{2}{\frac {a^{3}}{b^{3}}}+\ldots \right)\\&+{\frac {1}{2.3}}\left(2.3.\sigma {\frac {a}{b}}+3.4.\sigma _{1}{\frac {a^{2}}{b^{2}}}+4.5.\sigma _{2}{\frac {a^{3}}{b^{3}}}+\ldots \right)\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\\p^{2}=&{\frac {a^{2}}{b^{2}}}+2\left(\varpi {\frac {a^{2}}{b^{2}}}+\varpi _{1}{\frac {a^{3}}{b^{3}}}+\varpi _{2}{\frac {a^{4}}{b^{4}}}+\ldots \right)\\&+{\frac {2}{2}}\left(3\rho {\frac {a^{2}}{b^{2}}}+4\rho _{1}{\frac {a^{3}}{b^{3}}}+5\rho _{2}{\frac {a^{4}}{b^{4}}}+\ldots \right)\\&+{\frac {2}{2.3}}\left(3.4.\sigma {\frac {a^{2}}{b^{2}}}+4.5.\sigma _{1}{\frac {a^{3}}{b^{3}}}+5.6.\sigma _{2}{\frac {a^{4}}{b^{4}}}+\ldots \right)\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\\\\p^{3}=&{\frac {a^{3}}{b^{3}}}+3\left(\varpi {\frac {a^{3}}{b^{3}}}+\varpi _{1}{\frac {a^{4}}{b^{4}}}+\varpi _{2}{\frac {a^{5}}{b^{5}}}+\ldots \right)\\&+{\frac {3}{2}}\left(4\rho {\frac {a^{3}}{b^{3}}}+5\rho _{1}{\frac {a^{4}}{b^{4}}}+6\rho _{2}{\frac {a^{5}}{b^{5}}}+\ldots \right)\\&+{\frac {3}{2.3}}\left(4.5.\sigma {\frac {a^{3}}{b^{3}}}+5.6.\sigma _{1}{\frac {a^{4}}{b^{4}}}+6.7.\sigma _{2}{\frac {a^{5}}{b^{5}}}+\ldots \right)\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

et ainsi de suite.

11. Maintenant, puisqu’on a supposé

${\begin{aligned}\xi \,\ &=\varpi +\varpi _{1}x+\varpi _{2}x^{2}+\ldots ,\\\xi ^{2}&=\rho \ +\rho _{1}\,\ x+\rho _{2}\ x^{2}+\ldots ,\\\xi ^{3}&=\sigma \ +\sigma _{1}\ x+\sigma _{2}\ x^{2}+\ldots ,\\\end{aligned}}$