Page:Annales de mathématiques pures et appliquées, 1825-1826, Tome 16.djvu/156

 Les corrections sont expliquées en page de discussion

${\frac {\operatorname {d} V}{\operatorname {d} x}}Pq+Qr+Rs+\ldots \,;$

l’équation indéfinie se réduit donc à

${\frac {\operatorname {d} P}{\operatorname {d} x}}-{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{3}}}-\ldots =0,$

et donne, en intégrant

$P={\frac {\operatorname {d} Q}{\operatorname {d} x}}-{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}+\ldots +C\,;$

puis, en substituant

${\frac {\operatorname {d} V}{\operatorname {d} x}}=Qr+Rs+\ldots +q{\frac {\operatorname {d} Q}{\operatorname {d} x}}-q{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}+\ldots +Cq\,;$

et ensuite, en multipliant par $\operatorname {d} x$ et intégrant,

$V=\int Q\operatorname {d} q+\int R\operatorname {d} r+\ldots +\int q{\frac {\operatorname {d} Q}{\operatorname {d} x}}\operatorname {d} x-\int q{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}\operatorname {d} x+\ldots +Cp+C'\,;$

mais, nous avons trouvé ci-dessus, en intégrant par parties

${\begin{aligned}&\int Q\operatorname {d} q=Qq-p{\frac {\operatorname {d} Q}{\operatorname {d} x}}+\int p{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}\operatorname {d} x,\\\\&\int R\operatorname {d} r=Rr-q{\frac {\operatorname {d} R}{\operatorname {d} x}}+p{\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}-\int p{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{3}}}\operatorname {d} x,\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \,;\end{aligned}}$

il viendra donc, en substituant,