Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 7.djvu/242

on ait

$\mathrm {M_{2}+N_{2}} z+\ldots +{\frac {1}{t'}}\left(\mathrm {M_{2}^{(1)}+N_{2}^{(1)}} z+\ldots \right)+{\frac {1}{t'^{i-2}}}\mathrm {M} _{2}^{(i-2)}.$

et ainsi de suite. La formule (A) du n^o 5 donnera

${\begin{aligned}{\frac {1}{t^{i}}}=&\mathrm {M+N} z+\ldots \\&+{\frac {1}{t'}}\left(\mathrm {M^{(1)}+N^{(1)}} z+\ldots \right)\\&+{\frac {1}{t'^{2}}}\left(\mathrm {M^{(2)}+N^{(2)}} z+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \\&+{\frac {1}{t'^{i}}}\mathrm {M} ^{(i)}\\&+{\frac {1}{t}}\ \ \ \left\{{\begin{aligned}&\mathrm {M_{1}+N_{1}} z+\ldots \\+&{\frac {1}{t'}}\left(\mathrm {M_{1}^{(1)}+N_{1}^{(1)}} z+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \\+&{\frac {1}{t'^{i-1}}}\mathrm {M} _{1}^{(i-1)}\end{aligned}}\right\}\\\\&+{\frac {1}{t^{2}}}\ \ \left\{{\begin{aligned}&\mathrm {M_{2}+N_{2}} z+\ldots \\+&{\frac {1}{t'}}\left(\mathrm {M_{2}^{(1)}+N_{2}^{(1)}} z+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \\+&{\frac {1}{t'^{i-2}}}\mathrm {M} _{2}^{(i-2)}\end{aligned}}\right\}\\&+\ldots \ldots \ldots \ldots \ldots \\&+{\frac {1}{t^{n-1}}}\left\{{\begin{aligned}&\mathrm {M_{n-1}+N_{n-1}} z+\ldots \\+&{\frac {1}{t'}}\left(\mathrm {M_{n-1}^{(1)}+N_{n-1}^{(1)}} z+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \\+&{\frac {1}{t'^{i-n+1}}}\mathrm {M} _{n-1}^{(i-n+1)}\end{aligned}}\right\}\end{aligned}}$