Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/208

 Les corrections sont expliquées en page de discussion

$\Delta y'=b\Delta _{0}+\nu a\Delta _{0}+\lambda \left(c\Delta _{0}-b^{2}{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\right)-\mu ab{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}+b{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}+Z.$

Au moyen de ces valeurs, nous trouverons, en employant la transformation indiquée par l’équation (11),

$\iiint \Delta \operatorname {d} x'\operatorname {d} y'\operatorname {d} z'=\iiint \Delta \operatorname {d} a\operatorname {d} b\operatorname {d} c=\iiint \Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c+Z\iiint {\frac {\Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c$

$+\mu \iiint a{\frac {\Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c-\lambda \iiint b{\frac {\Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c\,;$

$\iiint \Delta x'\operatorname {d} x'\operatorname {d} y'\operatorname {d} z'=\iiint \Delta x'\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iiint a\Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c+Z\iiint a{\frac {\Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c$

$+\nu \iiint b\Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c-\mu \iiint \left(c\Delta _{0}-a^{2}{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\right)\operatorname {d} a\operatorname {d} b\operatorname {d} c-\lambda \iiint ab{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c\,;$

$\iiint \Delta y'\operatorname {d} x'\operatorname {d} y'\operatorname {d} z'=\iiint \Delta y'\operatorname {d} a\operatorname {d} b\operatorname {d} c=\iiint b\Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c+Z\iiint b{\frac {\Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c$

$-\nu \iiint a\Delta _{0}\operatorname {d} a\operatorname {d} b\operatorname {d} c+\lambda \iiint \left(c\Delta _{0}-b^{2}{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\right)\operatorname {d} a\operatorname {d} b\operatorname {d} c+\mu \iiint ab{\frac {\operatorname {d} \Delta _{0}}{\operatorname {d} c}}\operatorname {d} a\operatorname {d} b\operatorname {d} c.$

Maintenant on doit observer que $a,b,c,\Delta _{0}$ sont les valeurs de $x',y',z',\Delta$ qui auraient lieu dans le cas d’équilibre, et que, conséquemment, on doit avoir identiquement (10)

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