Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/13

Le texte de cette page a été corrigé et est conforme au fac-similé.

9

INDÉTERMINÉES.

\int \left\{\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y+\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y'''+\ldots \right\}\operatorname {d} x=0,

laquelle revient simplement à

0=\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y+\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y'''+\ldots

(I)

7. Cela posé, par la formule $(tu)'=ut'+tu',$ d’où $ut'=(tu)'-tu',$ on trouve facilement

{\begin{aligned}\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y\ \ =&\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y'\ =&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)Y\ \ \ \right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y''\,=&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)Y'\ \right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)'Y\ \ \right]'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''Y,\\\\\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y'''=&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)Y''\right]'-\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'Y'\right]'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)''Y\right]'-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''Y,\end{aligned}}

Au moyen de quoi l’équation (I) devient

\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''+\ldots \right]Y

+\left\{\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)''-\ldots \right]Y\right.

+\left.\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'+\ldots \right]Y'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)-\ldots \right]Y''+\ldots \right\}'=0\,;

(II)

or, tout ce qui suit la première ligne du premier membre de cette