Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/584

${\begin{aligned}d\zeta \ \ \,=&{\frac {z'(bd\varphi -a''d\psi )+z''(bd\psi -a'd\varphi )}{\alpha }},\\dx'\ =&{\frac {\xi d\psi -(bx'-a'x'')d\varpi }{\alpha }},\\dy'\ =&{\frac {\eta d\psi -(by'-a'y'')d\varpi }{\alpha }},\\dz'\ =&{\frac {\zeta d\psi -(bz'-a'z'')d\varpi }{\alpha }},\\dx''=&{\frac {\xi d\varphi -(a''x'-bx'')d\varpi }{\alpha }},\\dy''=&{\frac {\eta d\varphi -(a''y'-by'')d\varpi }{\alpha }},\\dz''=&{\frac {\zeta d\varphi -(a''z'-bz'')d\varpi }{\alpha }}.\end{aligned}}$

Si l’on différentie maintenant les valeurs de $x,y,z$ du Corollaire précédent, en supposant aussi constantes les quantités, $\beta ,b',b'',$ qu’on y substitue ensuite les valeurs de $d\xi ,d\eta ,d\zeta ,dx',dy',dz',\ldots$ qu’on vient de trouver, et qu’on fasse, pour abréger,

$d\Phi =bd\varphi -a''d\psi ,\quad d\Psi =bd\psi -a'd\varphi ,$

on aura

${\begin{aligned}dx=&{\frac {-\xi (b'd\Psi +b''d\Phi )+x'(\beta d\Phi -\alpha b'd\varpi )+x''(\beta d\Psi +\alpha b''d\varpi )}{\alpha ^{2}}},\\dy=&{\frac {-\eta (b'd\Psi +b''d\Phi )+y'(\beta d\Phi -\alpha b'd\varpi )+y''(\beta d\Psi +\alpha b''d\varpi )}{\alpha ^{2}}},\\dz=&{\frac {-\zeta (b'd\Psi +b''d\Phi )+z'(\beta d\Phi -\alpha b'd\varpi )+z''(\beta d\Psi +\alpha b''d\varpi )}{\alpha ^{2}}}.\end{aligned}}$

Et, si l’on substitue ces valeurs ainsi que celles de $x,y,z$ dans ces expressions $xdy-ydx,\ zdx-xdz,\ ydz-zdy,$ on aura, en employant les réductions du n^o 2,

${\begin{aligned}&xdy-ydx\\&=\zeta {\frac {\beta (a''b''-bb')d\Psi -\beta (a'b'-bb'')d\Phi +\alpha \left(a'b'^{2}+a''b''^{2}-2bb'b''\right)d\varpi }{\alpha ^{3}}}\\&\quad -z'{\frac {\left(b\beta ^{2}+\alpha b'b''\right)d\Phi +\left(a''\beta ^{2}+\alpha b'^{2}\right)d\Psi +\alpha \beta (a''b''-bb')d\varpi }{\alpha ^{3}}}\\&\quad +z''{\frac {\left(b\beta ^{2}+\alpha b'b''\right)d\Psi +\left(a'\beta ^{2}+\alpha b''^{2}\right)d\Phi -\alpha \beta (a'b'-bb'')d\varpi }{\alpha ^{3}}},\end{aligned}}$