Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 10.djvu/87

Si l’on développe le second membre de cette équation, par rapport aux puissances de ${\frac {1}{t_{1}}}-b,$ on aura

${\begin{aligned}&{\frac {1}{t^{x}t_{1}^{x_{1}}}}=\\&\left[\left({\frac {1}{t_{1}}}-b\right)^{x_{1}}+x_{1}b\left({\frac {1}{t_{1}}}-b\right)^{x_{1}-1}+{\frac {x_{1}(x_{1}-1)}{1.2}}b^{2}\left({\frac {1}{t_{1}}}-b\right)^{x_{1}-2}+\ldots \right]\\&\quad \times \left[a^{x}+x(c+ab){\frac {a^{x-1}}{{\cfrac {1}{t_{1}}}-b}}+{\frac {x(x-1)}{1.2}}(c+ab)^{2}{\frac {a^{x-2}}{\left({\cfrac {1}{t_{1}}}-b\right)^{2}}}+\ldots \right].\end{aligned}}$

Soient

${\begin{aligned}\mathrm {V} \quad =&a^{x},\\\mathrm {V} ^{(1)}=&x_{1}ba^{x}+x(c+ab)a^{x-1},\\\mathrm {V} ^{(2)}=&{\frac {x_{1}(x_{1}-1)}{1.2}}b^{2}a^{x}+x_{1}xb(c+ab)a^{x-1}+{\frac {x(x-1)}{1.2}}(c+ab)^{2}a^{x-2},\\\mathrm {V} ^{(3)}=&{\frac {x_{1}(x_{1}-1)(x_{1}-2)}{1.2.3}}b^{3}a^{x}+{\frac {x_{1}(x_{1}-1)}{1.2}}xb^{2}(c+ab)a^{x-1}\\&+x_{1}{\frac {x(x-1)}{1.2}}b(c+ab)^{2}a^{x-2}+{\frac {x(x-1)(x-2)}{1.2.3}}(c+ab)^{3}a^{x-3},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

on aura

${\frac {u}{t^{x}t^{x_{1}}}}=$

$u\times \left\{{\begin{aligned}&\mathrm {V} \left({\frac {1}{t_{1}}}-b\right)^{x_{1}}+\mathrm {V} ^{(1)}\left({\frac {1}{t_{1}}}-b\right)^{x_{1}-1}+\mathrm {V} ^{(2)}\left({\frac {1}{t_{1}}}-b\right)^{x_{1}-2}+\ldots \\&\qquad +\mathrm {V} ^{(x_{1})}+{\frac {\mathrm {V} ^{(x_{1}+1)}}{{\cfrac {1}{t_{1}}}-b}}+{\frac {\mathrm {V} ^{(x_{1}+2)}}{\left({\cfrac {1}{t_{1}}}-b\right)^{2}}}+\ldots +{\frac {\mathrm {V} ^{(x+x_{1})}}{\left({\cfrac {1}{t_{1}}}-b\right)^{x}}}\end{aligned}}\right\}\,;$

or l’équation

${\frac {1}{tt_{1}}}-{\frac {a}{t_{1}}}-{\frac {b}{t}}-c=0\quad {\text{donne}}\quad {\frac {1}{{\cfrac {1}{t_{1}}}-b^{x}}}={\frac {{\cfrac {1}{t}}-a}{c+ab}},$

partant

${\frac {u}{t^{x}t^{x_{1}}}}=$

$u\times \left\{{\begin{aligned}\mathrm {V} &\left({\frac {1}{t_{1}}}-b\right)^{x_{1}}+\mathrm {V} ^{(1)}\left({\frac {1}{t_{1}}}-b\right)^{x_{1}-1}+\ldots \\&+\mathrm {V} ^{(x_{1})}+{\frac {\mathrm {V} ^{(x_{1}+1)}}{c+ab}}\left({\frac {1}{t}}-a\right)+{\frac {\mathrm {V} ^{(x_{1}+2)}}{(c+ab)^{2}}}(\left({\frac {1}{t}}-a\right)^{2}+\ldots \\&+{\frac {\mathrm {V} ^{(x+x_{1})}}{(c+ab)^{x}}}\left({\frac {1}{t}}-a\right)^{x}\end{aligned}}\right\}.$