Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/127

équation de cette forme

${\begin{aligned}{\overset {1}{y}}_{x}+b_{q}{\overset {1}{y}}_{x-1}+^{1}\!b_{q}{\overset {1}{y}}_{x-2}+\ldots &=a_{q}\left({\overset {q}{y}}_{x}+\mathrm {A} {\overset {q}{y}}_{x-1}+^{1}\!\mathrm {A} {\overset {q}{y}}_{x-2}+\ldots \right)\\&+^{1}\!a_{q}\left({\overset {q}{y}}_{x-1}+\mathrm {A} {\overset {q}{y}}_{x-2}+^{1}\!\mathrm {A} {\overset {q}{y}}_{x-3}+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+c_{q}\left({\overset {q+1}{y}}_{x}+\mathrm {A} {\overset {q+1}{y}}_{x-1}+^{1}\!\mathrm {A} {\overset {q+1}{y}}_{x-2}+\ldots \right)\\&+^{1}\!c_{q}\left({\overset {q+1}{y}}_{x-1}+\mathrm {A} {\overset {q+1}{y}}_{x-2}+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \\&+{\overset {q}{u}}_{x}.\end{aligned}}$

Je substitue maintenant dans cette équation, au lieu de

${\overset {q}{y}}_{x}+\mathrm {A} {\overset {q}{y}}_{x-1}+\ldots ,\qquad {\overset {q}{y}}_{x-1}+\mathrm {A} {\overset {q}{y}}_{x-2}+\ldots ,$

leurs valeurs que donne la $q$^ième équation, ce qui produit la suivante

${\begin{aligned}{\overset {1}{y}}_{x}+b_{q}{\overset {1}{y}}_{x-1}+^{1}\!b_{q}{\overset {1}{y}}_{x-2}+\ldots &=a_{q}\left(\mathrm {B} {\overset {q+1}{y}}_{x}+^{1}\!\mathrm {B} {\overset {q+1}{y}}_{x-1}+^{2}\!\mathrm {B} {\overset {q+1}{y}}_{x-2}+\ldots \right)\\&+^{1}\!a_{q}\left(\mathrm {B} {\overset {q+1}{y}}_{x-1}+^{1}\!\mathrm {B} {\overset {q+1}{y}}_{x-2}+^{2}\!\mathrm {B} {\overset {q+1}{y}}_{x-3}+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+\ a_{q}\left(\mathrm {C} {\overset {q+2}{y}}_{x}+^{1}\!\mathrm {C} {\overset {q+2}{y}}_{x-1}+\ldots \right)\\&+^{1}\!a_{q}(\mathrm {C} {\overset {q+2}{y}}_{x-1}+\ldots )\\&+\ldots \ldots \ldots \ldots \\&+\ c_{q}\left({\overset {q+1}{y}}_{x}+\mathrm {A} {\overset {q+1}{y}}_{x-1}+\ldots \right)\\&+^{1}\!c_{q}({\overset {q+1}{y}}_{x-1}+\ldots )\\&+\ldots \ldots \ldots \ldots \\&+a_{q}\mathrm {X} _{x}+^{1}\!a_{q}\mathrm {X} _{x-1}+\ldots \\&+{\overset {q}{u}}_{x}\,;\end{aligned}}$