Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/69

65

INDÉTERMINÉES.

$+\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)''-\ldots \right]X\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'+\ldots \right]X'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)-\ldots \right]X''+\ldots \\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)''-\ldots \right]Y\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'+\ldots \right]Y'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)-\ldots \right]Y''+\ldots \\\\\end{aligned}}\right\}\,;$

équation qui se décompose en ces deux-ci :

$0=\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'''+\ldots \right]X\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)''-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'''+\ldots \right]Y\end{aligned}}\right\}(4)$

$Const.=\left\{{\begin{aligned}&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)''-\ldots \right]X\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)'+\ldots \right]X'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} x'''}}\right)-\ldots \right]X''+\ldots \\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)'+\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)''-\ldots \right]Y\\\\+&\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)-\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)'+\ldots \right]Y'+\left[\left({\frac {\operatorname {d} V}{\operatorname {d} y'''}}\right)-\ldots \right]Y''+\ldots \\\\\end{aligned}}\right\}(5)$

61. Cela posé, les fonctions arbitraires $X$ et $Y$ de $t$ devant conserver l’indépendance la plus entière dans toute l’étendue de l’intégrale, l’équation (4) donnera séparément