Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/30

26

INTÉGRALES

$\int \left\{\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X+\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X''+\ldots \right.$

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

laquelle revient simplement à

$\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)X+\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)X'+\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)X''+\ldots$

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

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

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

 

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