![{\displaystyle \left.{\begin{aligned}&\left({\frac {a}{r}}\right)^{2}+\left({\frac {a'}{r'}}\right)^{2}+\left({\frac {a''}{r''}}\right)^{2}=1,\\&\qquad \qquad \left({\frac {b}{r}}\right)\left({\frac {c}{r}}\right)+\left({\frac {b'}{r'}}\right)\left({\frac {c'}{r'}}\right)+\left({\frac {b''}{r''}}\right)\left({\frac {c''}{r''}}\right)=0,\\\\&\left({\frac {b}{r}}\right)^{2}+\left({\frac {b'}{r'}}\right)^{2}+\left({\frac {b''}{r''}}\right)^{2}=1,\\&\qquad \qquad \left({\frac {c}{r}}\right)\left({\frac {a}{r}}\right)+\left({\frac {c'}{r'}}\right)\left({\frac {a'}{r'}}\right)+\left({\frac {c''}{r''}}\right)\left({\frac {a''}{r''}}\right)=0,\\\\&\left({\frac {c}{r}}\right)^{2}+\left({\frac {c'}{r'}}\right)^{2}+\left({\frac {c''}{r''}}\right)^{2}=1,\\&\qquad \qquad \left({\frac {a}{r}}\right)\left({\frac {b}{r}}\right)+\left({\frac {a'}{r'}}\right)\left({\frac {b'}{r'}}\right)+\left({\frac {a''}{r''}}\right)\left({\frac {b''}{r''}}\right)=0\,;\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5b599745ddc3f2275652dce98b259860689557fb)
(10)
![{\displaystyle \left.{\begin{aligned}&\left(p^{2}+q^{2}+r^{2}-1\right)t^{2}+2(p'p''+q'q''+r'r'')uv\\\\+&\left(p'^{2}+q'^{2}+r'^{2}-1\right)u^{2}+2(p''p+q''q+r''r)\tau \\\\+&\left(p''^{2}+q''^{2}+r''^{2}-1\right)v^{2}+2(pp'+qq'+rr')tu\end{aligned}}\right\}=0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c73fce2041b9bc42ad13fa4886218c7d210bb45)
équation qui x parce qu’elle doit être identique, donne
![{\displaystyle \left.{\begin{array}{ll}p^{2}+q^{2}+r^{2}=1,&p'p''+q'q''+r'r''=0,\\\\p'^{2}+q'^{2}+r'^{2}=1,&p''p+q''q+r''r=0,\\\\p''^{2}+q''^{2}+r''^{2}=1,&pp'+qq'+rr'=0.\end{array}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7908fb2385cbafabf0926ffaf0a0d997b34366a7)
(3)
D’un autre côté, si l’on prend, tour à tour, la somme des produits respectifs des équations (2), d’abord par
, ensuite par
puig enfin par
en ayant égard aux relations (3), il viendra,
![{\displaystyle t=px+qy+rz,\quad u=p'x+q'y+r'z,\quad v=p''x+q''y+r''z\,;\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5d47e36326031720689fce9ebc0238106c3b972)
(4)
substituant dans (1), transposant et développant, on aura
![{\displaystyle \left.{\begin{aligned}&\left(p^{2}+p'^{2}+p''^{2}-1\right)x^{2}+2(qr+q'r'+q''r'')yz\\\\+&\left(q^{2}+q'^{2}+q''^{2}-1\right)y^{2}+2(rp+r'p'+r''p'')zx\\\\+&\left(r^{2}+r'^{2}+r''^{2}-1\right)z^{2}+2(pq+p'q'+p''q'')xy\end{aligned}}\right\}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1490026a5a9316f52c7d7fe6f521301fb48139)
équation qui, devant aussi être identique, donne