Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/209

 Les corrections sont expliquées en page de discussion

${\begin{aligned}&\iiint b\Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c=0\,;&\end{aligned}}$

ayant donc égard à ces conditions et posant, pour abréger,

${\begin{aligned}&\iiint {\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iint (\Delta ''-\Delta ')\operatorname {d} a\operatorname {d} b=L,\\\\&\iiint a{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iint (\Delta ''-\Delta ')a\operatorname {d} a\operatorname {d} b=M,\\\\&\iiint b{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iint (\Delta ''-\Delta ')b\operatorname {d} a\operatorname {d} b=N,\\\\&\iiint a^{2}{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iint (\Delta ''-\Delta ')a^{2}\operatorname {d} a\operatorname {d} b=P,\\\\&\iiint b^{2}{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iint (\Delta ''-\Delta ')b^{2}\operatorname {d} a\operatorname {d} b=Q,\\\\&\iiint ab{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iint (\Delta ''-\Delta ')ab\operatorname {d} a\operatorname {d} b=R,\\\\&\qquad \qquad \iiint c\Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c=MV\,;\end{aligned}}$

nous trouverons

${\begin{aligned}&\iiint \Delta \operatorname {d} x'\operatorname {d} y'\operatorname {d} z'=m+LZ+\mu M-\lambda N,\\\\&\iiint \Delta x'\operatorname {d} x'\operatorname {d} y'\operatorname {d} z'=MZ-\mu (mV-P)-\lambda R,\\\\&\iiint \Delta y'\operatorname {d} x'\operatorname {d} y'\operatorname {d} z'=NZ+\lambda (mV-Q)+\mu R\,;\end{aligned}}$

équations dans lesquelles $L,M,N,P,Q,R,V$ expriment des