![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f353261c4c2c48b473fdcdbe4d0edf55d3e563d)
Si l’on différentie maintenant les valeurs de
du Corollaire précédent, en supposant aussi constantes les quantités,
qu’on y substitue ensuite les valeurs de
qu’on vient de trouver, et qu’on fasse, pour abréger,
![{\displaystyle d\Phi =bd\varphi -a''d\psi ,\quad d\Psi =bd\psi -a'd\varphi ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d27cb0e074e7ca15c86adeb93bd61ab1a0a11c70)
on aura
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2715264048698d765b1f2f71c2bc171c6dfbc2)
Et, si l’on substitue ces valeurs ainsi que celles de
dans ces expressions
on aura, en employant les réductions du no 2,
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af38fc9befd8880f94323faae9f9d96a29660f5c)