Page:Mémoires de l’Académie des sciences, Tome 22.djvu/239

on trouvera, en posant $\operatorname {F} \left({\frac {\alpha }{\beta }},1\right)=\Phi \left({\frac {\alpha }{\beta }}\right),$

$\operatorname {\frac {1}{F(\alpha ,\beta )}} ={\frac {1}{\Phi '(\theta _{0})}}{\frac {1}{\alpha -\theta _{0}\beta }}+\ldots +{\frac {1}{\Phi '(\theta _{m-1})}}{\frac {1}{\alpha -\theta _{m-1}\beta }},$

et, par suite, l’intégrale de l’équation

(26)

$\operatorname {F} (\operatorname {D} _{x},\operatorname {D} _{y})u=f(x,y),$

sera

(27)

$u={\frac {f(x,y)}{\operatorname {F} (\operatorname {D} _{x},\operatorname {D} _{y})}}$

${\begin{aligned}={\frac {1}{\Phi '(\theta _{0})}}{\frac {f(x,y)}{\operatorname {D} _{x}-\theta _{0}\operatorname {D} _{y}}}+{\frac {1}{\Phi '(\theta _{1})}}{\frac {f(x,y)}{\operatorname {D} _{x}-\theta _{1}\operatorname {D} _{y}}}+\ldots +{\frac {1}{\Phi '(\theta _{m-1})}}{\frac {f(x,y)}{\operatorname {D} _{x}-\theta _{m-1}\operatorname {D} _{y}}}.\end{aligned}}$

En général, si $\operatorname {F} (\alpha ,\beta ,\gamma ,\ldots \theta )$ se décompose en plusieurs facteurs du premier degré, ou de la forme

$a\alpha +b\beta +c\gamma +\ldots +k\theta +l,$

alors, en posant

(28)

${\frac {d\operatorname {F} (\alpha ,\beta ,\gamma ,\ldots \theta )}{d\alpha }}=\varphi (\alpha ,\beta ,\gamma ,\ldots \theta ),$

on trouvera

(29)

${\begin{aligned}&{\frac {1}{\operatorname {F} (\alpha ,\beta ,\gamma ,\ldots \theta )}}=\\&\qquad {\frac {1}{\varphi \left(-{\cfrac {b\beta +\beta \gamma \ldots +k\theta +l}{a}},\beta ,\gamma \ldots \theta \right)}}{\frac {1}{a\alpha +b\beta \ldots +k\theta +l}}\end{aligned}}$

+ etc…,

et, par suite, l’intégrale générale de l’équation

(30)

$\operatorname {F} (\operatorname {D} _{x},\operatorname {D} _{y},\operatorname {D} _{z}\ldots \operatorname {D} _{t})u=f(x,y,z\ldots t)$

sera

(31)

$u={\frac {f(x,y,z\ldots t)}{\operatorname {F} (\operatorname {D} _{x},\operatorname {D} _{y}\ldots \operatorname {D} _{t})}}$

$={\frac {f(x,y,z\ldots t)}{\begin{aligned}&(a\operatorname {D} _{x}+b\operatorname {D} _{y}\ldots +k\operatorname {D} _{1}+l)\\&\times \varphi \left[-\left({\cfrac {b}{a}}\operatorname {D} _{x}+{\cfrac {c}{a}}\operatorname {D} _{y}\ldots +{\cfrac {k}{a}}\operatorname {D} _{1}+{\cfrac {l}{a}}\right),\operatorname {D} _{y},\operatorname {D} _{z}\ldots \operatorname {D} _{1}\right]\end{aligned}}}$

$+$ etc…,