d’où l’on conclura, par l’Article VII,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97d37de5ad7363d52632eb5d2659ae78d0870203)
et ainsi de suite, partant
![{\displaystyle {\overset {n-1}{u}}_{x+1}=^{n-1}\!z_{x+1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18b8d28b74c7e06007aecea1e281e7727f365a7f)
![{\displaystyle =\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}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ba754c8dfe0cf286a2f7d8c7fcc3a9635c2ced3)
pareillement
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c7d8e70a2fdead060f0351b9ac761f6e0ba7e48)
d’où l’on conclura, en substituant ces valeurs dans la formule (II) de l’article VII et faisant
pour abréger,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efbcd725b026884344b004344dfe1a7b4ffd5f82)
Si
on fera
Soit
et l’on aura
![{\displaystyle y_{x}=\varphi _{1}\varphi _{2}\ldots \varphi _{x}p^{x+n-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d6f7f849bf26afbe0496de03cf8b218ed5d9ed3)
![{\displaystyle \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\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/008626495d4a71d78f5a5817159c88c9eec4d668)
![{\displaystyle +{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/974c2e5b63ed9a88743d6eb8c2a6070b32e5e2bb)
et
étant deux constantes arbitraires.