et faisant, pour abréger,
![{\displaystyle {\begin{alignedat}{2}\alpha \ \ =&a'a''-b^{2},&\beta \ \ =&b'b''-ab,\\\alpha '\ =&a\ a''-b'^{2},&\beta '\ =&b\ b''-a'b',\\\alpha ''=&a\ a'\ -b''^{2},\qquad &\beta ''=&b\ b'\ -a''b'',\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33947d66d9dd537654657707e28e6ffd1e7409cd)
on aura de même
![{\displaystyle {\begin{alignedat}{2}\xi ^{2}\ \,+\eta ^{2}\,\ +\zeta ^{2}\ \ =&\alpha ,&\xi '\xi ''+\eta '\eta ''+\zeta '\zeta ''=&\beta ,\\\xi '^{2}\ +\eta '^{2}\,+\zeta '^{2}\ =&\alpha ',&\xi \ \xi ''+\eta \ \eta ''+\zeta \ \zeta ''=&\beta ',\\\xi ''^{2}+\eta ''^{2}+\zeta ''^{2}=&\alpha '',&\qquad \xi \ \xi '\ +\eta \ \eta '\ +\zeta \,\zeta '\ =&\beta ''.\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71b457338c523a486547ecb72261f8cbe174da32)
C’est ce qu’il est aisé de vérifier par la substitution des valeurs de ![{\displaystyle \xi ,\xi '\ldots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2421de0ab2d88779740b44396503bfb9df381999)
en ![{\displaystyle x,x',\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17693068ffbdf4e6ffd54618cd8e3893a463e2f7)
2. Donc, si l’on fait pareillement
![{\displaystyle {\begin{alignedat}{3}\mathrm {X} \ \ =&\eta '\zeta ''-\zeta '\eta '',\qquad &\mathrm {Y} \ \ =&\zeta '\xi ''-\xi '\zeta '',\qquad &\mathrm {Z} \ \ =&\xi '\eta ''-\eta '\xi '',\\\mathrm {X} '\ =&\eta ''\zeta \,\ -\zeta ''\eta ,&\mathrm {Y} '\ =&\zeta ''\xi \ -\xi ''\zeta ,&\mathrm {Z} '\ =&\xi ''\eta \ -\eta ''\xi ,\\\mathrm {X} ''=&\eta \ \zeta '\ -\zeta \eta ',&\mathrm {Y} ''=&\zeta \ \xi '\ -\xi \zeta ',&\mathrm {Z} ''=&\xi \ \eta '\ -\eta \xi ',\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f117f19f260ec3bddde1b15da9176e0ab2eb8bd)
et ensuite
![{\displaystyle {\begin{alignedat}{2}\mathrm {A} \ \ =&\alpha '\alpha ''-\beta ^{2},\qquad &\mathrm {B} \ \ =&\beta '\beta ''-\alpha \beta ,\\\mathrm {A} '\ =&\alpha \ \alpha ''-\beta '^{2},&\mathrm {B} '\ =&\beta \ \beta ''-\alpha '\beta ',\\\mathrm {A} ''=&\alpha \ \alpha '\ -\beta ''^{2},&\mathrm {B} ''=&\beta \ \beta '\ -\alpha ''\beta '',\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b14c34f73358c33c8fec707c1f58fb1947a232be)
on aura aussi
![{\displaystyle {\begin{alignedat}{2}\mathrm {X^{2}\ \,+Y^{2}\ +Z^{2}} \ \ =&\mathrm {A} ,\qquad &\mathrm {X'X''+Y'Y''+Z'Z''} =&\mathrm {B} ,\\\mathrm {X'^{2}\,+Y'^{2}\,+Z'^{2}} \ =&\mathrm {A} ',&\mathrm {X\,X''\ +YY''\ +ZZ''} \ =&\mathrm {B} ',\\\mathrm {X''^{2}+Y''^{2}+Z''^{2}} =&\mathrm {A} '',&\mathrm {X\,X'\,\ +YY'\ \,+ZZ'} \,\ =&\mathrm {B} '',\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92e626141047b3a268cd7f4c632b8e4d0099c7a8)
3. Or en substituant les valeurs de
en
et faisant, pour abréger,
![{\displaystyle \Delta =xy'z''+yz'x''+zx'y''-xz'y''-yx'z''-zy'x'',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0b2fa7eb15621567adf67735b675206b14dc919)
on trouve
![{\displaystyle {\begin{alignedat}{3}\mathrm {X} \ \ =&\delta x,\qquad &\mathrm {Y} \ \ =&\delta y,\qquad &\mathrm {Z} \ \ =&\delta z,\\\mathrm {X} '\ =&\delta x',&\mathrm {Y} '\ =&\delta y',&\mathrm {Z} '\ =&\delta z',\\\mathrm {X} ''=&\delta x'',&\mathrm {Y} ''=&\delta y'',&\mathrm {Z} ''=&\delta z'',\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e5a0cee9e3fb95cfe992becbda3e4be85896d7)