Page:Joseph Louis de Lagrange - Œuvres, Tome 3.djvu/664

et faisant, pour abréger,

${\begin{alignedat}{2}\alpha \ \ =&a'a''-b^{2},&\beta \ \ =&b'b''-ab,\\\alpha '\ =&a\ a''-b'^{2},&\beta '\ =&b\ b''-a'b',\\\alpha ''=&a\ a'\ -b''^{2},\qquad &\beta ''=&b\ b'\ -a''b'',\end{alignedat}}$

on aura de même

${\begin{alignedat}{2}\xi ^{2}\ \,+\eta ^{2}\,\ +\zeta ^{2}\ \ =&\alpha ,&\xi '\xi ''+\eta '\eta ''+\zeta '\zeta ''=&\beta ,\\\xi '^{2}\ +\eta '^{2}\,+\zeta '^{2}\ =&\alpha ',&\xi \ \xi ''+\eta \ \eta ''+\zeta \ \zeta ''=&\beta ',\\\xi ''^{2}+\eta ''^{2}+\zeta ''^{2}=&\alpha '',&\qquad \xi \ \xi '\ +\eta \ \eta '\ +\zeta \,\zeta '\ =&\beta ''.\end{alignedat}}$

C’est ce qu’il est aisé de vérifier par la substitution des valeurs de $\xi ,\xi '\ldots ,$$\alpha ,\alpha '\ldots$ en $x,x',\ldots .$

2. Donc, si l’on fait pareillement

${\begin{alignedat}{3}\mathrm {X} \ \ =&\eta '\zeta ''-\zeta '\eta '',\qquad &\mathrm {Y} \ \ =&\zeta '\xi ''-\xi '\zeta '',\qquad &\mathrm {Z} \ \ =&\xi '\eta ''-\eta '\xi '',\\\mathrm {X} '\ =&\eta ''\zeta \,\ -\zeta ''\eta ,&\mathrm {Y} '\ =&\zeta ''\xi \ -\xi ''\zeta ,&\mathrm {Z} '\ =&\xi ''\eta \ -\eta ''\xi ,\\\mathrm {X} ''=&\eta \ \zeta '\ -\zeta \eta ',&\mathrm {Y} ''=&\zeta \ \xi '\ -\xi \zeta ',&\mathrm {Z} ''=&\xi \ \eta '\ -\eta \xi ',\end{alignedat}}$

et ensuite

${\begin{alignedat}{2}\mathrm {A} \ \ =&\alpha '\alpha ''-\beta ^{2},\qquad &\mathrm {B} \ \ =&\beta '\beta ''-\alpha \beta ,\\\mathrm {A} '\ =&\alpha \ \alpha ''-\beta '^{2},&\mathrm {B} '\ =&\beta \ \beta ''-\alpha '\beta ',\\\mathrm {A} ''=&\alpha \ \alpha '\ -\beta ''^{2},&\mathrm {B} ''=&\beta \ \beta '\ -\alpha ''\beta '',\end{alignedat}}$

on aura aussi

${\begin{alignedat}{2}\mathrm {X^{2}\ \,+Y^{2}\ +Z^{2}} \ \ =&\mathrm {A} ,\qquad &\mathrm {X'X''+Y'Y''+Z'Z''} =&\mathrm {B} ,\\\mathrm {X'^{2}\,+Y'^{2}\,+Z'^{2}} \ =&\mathrm {A} ',&\mathrm {X\,X''\ +YY''\ +ZZ''} \ =&\mathrm {B} ',\\\mathrm {X''^{2}+Y''^{2}+Z''^{2}} =&\mathrm {A} '',&\mathrm {X\,X'\,\ +YY'\ \,+ZZ'} \,\ =&\mathrm {B} '',\end{alignedat}}$

3. Or en substituant les valeurs de $\xi ,\xi ',\ldots$ en $x,x',\ldots$ et faisant, pour abréger,

$\Delta =xy'z''+yz'x''+zx'y''-xz'y''-yx'z''-zy'x'',$

on trouve

${\begin{alignedat}{3}\mathrm {X} \ \ =&\delta x,\qquad &\mathrm {Y} \ \ =&\delta y,\qquad &\mathrm {Z} \ \ =&\delta z,\\\mathrm {X} '\ =&\delta x',&\mathrm {Y} '\ =&\delta y',&\mathrm {Z} '\ =&\delta z',\\\mathrm {X} ''=&\delta x'',&\mathrm {Y} ''=&\delta y'',&\mathrm {Z} ''=&\delta z'',\\\end{alignedat}}$