et faisant
![{\displaystyle {\begin{alignedat}{2}\mathrm {M} \ \,=&{\frac {\alpha +\mu }{2}},\qquad &\mathrm {N} \ \,=&{\frac {\beta +\nu }{2}},\\\mathrm {M} '=&{\frac {\alpha -\mu }{2}},&\mathrm {N} '=&{\frac {\beta -\nu }{2}},\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a318f0a67bfad7785f5ce549dd0df971b574efb0)
on aura
![{\displaystyle {\begin{array}{ll}\alpha =\mathrm {A} ,&\alpha ^{2}-\mu ^{2}+4\beta =4\mathrm {B} ,\\\alpha \beta -\mu \nu =2\mathrm {C} ,&\beta ^{2}-\nu ^{2}=4\mathrm {D} ,\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e70319a772ebe4ebde0258b4828319cb32453b7)
d’où l’on tire d’abord
![{\displaystyle \alpha =\mathrm {A} ,\quad \beta =\mathrm {\frac {4B-A^{2}+\mu ^{2}}{4}} \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/361b695a57fc836521ae10b1eb7b3a8351526f69)
substituant ensuite ces valeurs dans les deux dernières équations, on aura
![{\displaystyle {\begin{aligned}&\mathrm {A\mu ^{2}-4\mu \nu -A^{3}+4AB-8C} =0,\\&\mu ^{4}+2\mathrm {\left(4B-A^{2}\right)\mu ^{2}-16\nu ^{2}+\left(4B-A^{2}\right)^{2}-64D} =0.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ae8b0fe21de56063d34aa7e6074cb396bc7a267)
On fera maintenant
![{\displaystyle u=a\mu +b\nu ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ae34a9dca4280f0b97e9b20aa4dc5c57fe4d4ff)
et substituant, par exemple,
à la place de
on aura ces deux équations-ci
![{\displaystyle {\begin{aligned}&\left(\mathrm {A} -{\frac {4a}{b}}\right)\mu ^{2}-{\frac {4\mu u}{b}}-\mathrm {A^{3}+4AB-8C} =0,\\&\mu ^{4}+\left(\mathrm {8B-2A^{2}} -{\frac {16a^{2}}{b^{2}}}\right)\mu ^{2}+{\frac {32a\mu u}{b^{2}}}-{\frac {16u^{2}}{b^{2}}}+\mathrm {\left(4B-A^{2}\right)^{2}-64D} =0\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af13f15516f93777c0e9f451647c2cb1fe9d1f7e)
d’où lon chassera
pour avoir une réduite en ![{\displaystyle u.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/edd5636410da69bac33da075162221527401793c)
Supposons, pour abréger,
![{\displaystyle {\begin{aligned}&\mathrm {\left(A^{3}+4AB-8C\right)=F} ,\\&\mathrm {\left(4B-A^{2}\right)^{2}-64D=G} ,\\&b\mathrm {A} -4a=f,\\&8b^{2}\mathrm {B} -2b^{2}\mathrm {A} ^{2}-16a^{2}=g,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f65dc146312f65c37bba999bd57379ee25d0e3)
on aura
![{\displaystyle {\begin{aligned}&f\mu ^{2}-4\mu u-b\mathrm {F} =0,\\&b^{2}\mu ^{4}+g\mu ^{2}+32a\mu u-16u^{2}+b^{2}\mathrm {G} =0\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed396c4c522b97939657db55069ef9cd1c0d10ab)