Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/107

d’où l’on conclura, par l’Article VII,

${\begin{aligned}{\overset {1}{u}}_{x}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x}p^{x}\Delta {\frac {^{1}\!p^{x-1}}{p^{x-1}}}=\varphi _{1}\varphi _{2}\ldots \varphi _{x}\left(^{1}\!p-p\right)\,^{1}\!p^{x-1},\\^{1}{\overset {1}{u}}_{x}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x}\left(^{2}\!p-p\right)\,^{2}\!p^{x-1},\\^{2}{\overset {1}{u}}_{x}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x}\left(^{3}\!p-p\right)\,^{3}\!p^{x-1},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots ,\\\!{\overset {2}{u}}_{x}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x}\left(^{2}\!p-p\right)\left(^{2}\!p-^{1}\!p\right)\,^{2}\!p^{x-2},\\^{1}\!{\overset {2}{u}}_{x}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x}\left(^{3}\!p-p\right)\left(^{3}\!p-^{1}\!p\right)\,^{3}\!p^{x-2},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots ,\\\!\!{\overset {3}{u}}_{x}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x}\left(^{3}\!p-p\right)\left(^{3}\!p-^{1}\!p\right)\left(^{3}\!p-^{2}\!p\right)\,^{3}\!p^{x-3},\\\ldots &\ldots \ldots \ldots \ldots \ldots \ldots ,\end{aligned}}$

et ainsi de suite, partant

${\overset {n-1}{u}}_{x+1}=^{n-1}\!z_{x+1}$

$=\varphi _{1}\varphi _{2}\ldots \varphi _{x+1}\left(^{n-1}\!p-p\right)\left(^{n-1}\!p-^{1}\!p\right)\left(^{n-1}\!p-^{2}\!p\right)\ldots ^{n-1}\!p^{x-n+2}\,;$

pareillement

${\begin{aligned}^{n-2}\!z_{x+1}=&\varphi _{1}\varphi _{2}\ldots \varphi _{x+1}\left(^{n-2}\!p-p\right)\left(^{n-2}\!p-^{1}\!p\right)\ldots ^{n-2}\!p^{x-n+2},\\\ldots \ldots &\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

d’où l’on conclura, en substituant ces valeurs dans la formule (II) de l’article VII et faisant $X_{x}=\varphi _{1}\varphi _{2}\ldots \varphi _{x}\,^{1}\!X_{x}$ pour abréger,

${\begin{aligned}y_{x}&={\frac {\varphi _{1}\varphi _{2}\ldots \varphi _{x}}{\left(p-^{1}\!p\right)\left(p-^{2}\!p\right)\left(p-^{3}\!p\right)\ldots }}p^{x+n-1}\left(\ \mathrm {G} +\sum {\frac {^{1}\!\mathrm {X} _{x+1}}{p^{x+1}}}\right)\\&+{\frac {\varphi _{1}\varphi _{2}\ldots \varphi _{x}}{\left(^{1}\!p-p\right)\left(^{1}\!p-^{2}\!p\right)\ldots }}\qquad \quad ^{1}\!p^{x+n-1}\left(^{1}\!\mathrm {G} +\sum {\frac {^{1}\!\mathrm {X} _{x+1}}{^{1}\!p^{x+1}}}\right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .\end{aligned}}$

Si $p=^{1}\!p,$ on fera $^{1}\!p=p+dp.$ Soit $\mathrm {K} ={\frac {1}{\left(p-^{2}\!p\right)\left(p-^{3}\!p\right)\ldots }},$ et l’on aura

$y_{x}=\varphi _{1}\varphi _{2}\ldots \varphi _{x}p^{x+n-1}$

$\times \left\{\mathrm {B} +\mathrm {D} x-{\frac {\mathrm {K} }{p}}\sum {\frac {^{1}\!\mathrm {X} _{x+1}}{p^{x+1}}}(x+1)+\left[{\frac {d\mathrm {K} }{dp}}+{\frac {\mathrm {K} }{p}}(x+n-1)\right]\sum {\frac {^{1}\!\mathrm {X} _{x+1}}{p^{x+1}}}\right\}$

$+{\frac {\varphi _{1}\varphi _{2}\ldots \varphi _{x}}{\left(^{2}\!p-p\right)^{2}\left(^{2}\!p-^{3}\!p\right)\ldots }}\,^{2}\!p^{x+n-1}\left(^{2}\!\mathrm {G} +\sum {\frac {^{1}\!\mathrm {X} _{x+1}}{^{2}\!p^{x+1}}}\right),$

$\mathrm {B}$ et $\mathrm {D}$ étant deux constantes arbitraires.