Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 8.djvu/21

ce qui donnera, en combinant ces équations avec l’équation (3) du problème,

${\begin{aligned}\alpha _{n}.&\sideset {_{n-1}}{_{x}}y+\beta _{n}.\sideset {_{n-1}}{_{x-1}}y+\gamma _{n}.\sideset {_{n-1}}{_{x-2}}y+\ldots \\&=a_{n-1}\left(\alpha _{n}.\sideset {_{n-1}}{_{x-1}}y+\beta _{n}.\sideset {_{n-1}}{_{x-2}}y+\ldots \right)\\&=b_{n-1}\left(\alpha _{n}.\sideset {_{n-1}}{_{x-2}}y+\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \\&+h_{n-1}\left(\sideset {_{n}}{_{x}}y+\mathrm {A} _{n}.\sideset {_{n}}{_{x-1}}y+\mathrm {B} _{n}.\sideset {_{n}}{_{x-2}}y+\ldots +\right.\\&\qquad \qquad \qquad \left.-\mathrm {H} _{n}.\sideset {_{n-1}}{_{x}}y-\mathrm {M} _{n}.\sideset {_{n-1}}{_{x-1}}y-\ldots \right)\\&+l_{n-1}\left(\sideset {_{n}}{_{x-1}}y+\mathrm {A} _{n}.\sideset {_{n}}{_{x-1}}y+\mathrm {B} _{n}.\sideset {_{n}}{_{x-2}}y+\ldots +\right.\\&\qquad \qquad \qquad \left.-\mathrm {H} _{n}.\sideset {_{n-1}}{_{x-1}}y-\mathrm {M} _{n}.\sideset {_{n-1}}{_{x-2}}y-\ldots \right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&+u_{n-1}\left(\alpha _{n}+\beta _{n}+\gamma _{n}+\ldots \right)\\&+\mathrm {N} _{n}(h_{n-1}+l_{n-1}+\ldots ).\end{aligned}}$

Donc

${\begin{aligned}&\sideset {_{n}}{_{x}}y=\sideset {_{n}}{_{x-1}}y\left(-\mathrm {A} _{n}-{\frac {l_{n-1}}{h_{n-1}}}\right)+\sideset {_{n}}{_{x-2}}y\left(-\mathrm {B} _{n}-{\frac {l_{n-1}\mathrm {A} _{n}}{h_{n-1}}}-{\frac {p_{n-1}}{h_{n-1}}}\right)+\ldots \\&+\sideset {_{n-1}}{_{x}}y\left({\frac {\alpha _{n}}{h_{n-1}}}+\mathrm {H} _{n}\right)+\sideset {_{n-1}}{_{x-1}}y\left({\frac {\beta _{n}}{h_{n-1}}}+\mathrm {M} _{n}+l_{n-1}{\frac {\mathrm {H} _{n}}{h_{n-1}}}-{\frac {\alpha _{n}a_{n-1}}{h_{n-1}}}\right)\\&+\sideset {_{n-1}}{_{x-2}}y\left({\frac {\gamma _{n}}{h_{n-1}}}+\mathrm {P} _{n}+{\frac {l_{n-1}\mathrm {M} _{n}}{h_{n-1}}}+{\frac {p_{n-1}\mathrm {H} _{n}}{h_{n-1}}}-{\frac {\alpha _{n}b_{n-1}}{h_{n-1}}}-{\frac {a_{n-1}\beta _{n}}{h_{n-1}}}\right)\\&+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&-u_{n-1}{\frac {\alpha _{n}+\beta _{n}+\ldots }{h_{n-1}}}-\mathrm {N} _{n}{\frac {h_{n-1}+l_{n-1}+\ldots }{h_{n-1}}}\,;\end{aligned}}$

d’où l’on tirera, en comparant avec l’équation (4),

${\begin{array}{ll}(\mathrm {I} )&{\frac {\alpha _{n}}{h_{n-1}}}+\mathrm {H} _{n}=h_{n},\\(\mathrm {II} )&a_{n}=-\mathrm {A} _{n}-{\frac {l_{n-1}}{h_{n-1}}},\\(\mathrm {III} )&l_{n}={\frac {\beta _{n}}{h_{n-1}}}+\mathrm {M} _{n}+{\frac {l_{n-1}\mathrm {H} _{n}}{h_{n-1}}}-{\frac {\alpha _{n}a_{n-1}}{h_{n-1}}},\\(\mathrm {IV} )&b_{n}=-\mathrm {B} _{n}-{\frac {l_{n-1}\mathrm {A} _{n}}{h_{n-1}}}-{\frac {p_{n-1}}{h_{n-1}}},\\(\mathrm {V} )&p_{n}={\frac {\gamma _{n}}{h_{n-1}}}+\mathrm {P} _{n}+{\frac {l_{n-1}\mathrm {M} _{n}}{h_{n-1}}}+{\frac {p_{n-1}\mathrm {H} _{n}}{h_{n-1}}}-{\frac {\alpha _{n}b_{n-1}}{h_{n-1}}}-{\frac {a_{n-1}\beta _{n}}{h_{n-1}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \end{array}}$