Page:Annales de mathématiques pures et appliquées, 1827-1828, Tome 18.djvu/239

 Les corrections sont expliquées en page de discussion

$\left.{\begin{aligned}&p^{2}+p'^{2}+p''^{2}=1,\\&q^{2}+q'^{2}+q''^{2}=1,\\&r^{2}+r'^{2}+r''^{2}=1\,;\end{aligned}}\right\}$(3)$\qquad \left.{\begin{aligned}&qr+q'r'+q''r''=0,\\&rp+r'p'+r''p''=0,\\&pq+p'q'+p''q''=0\,;\end{aligned}}\right\}$(4)

ou, ce qui revient au même, par celles-ci

$\left.{\begin{aligned}&p^{2}+q^{2}+r^{2}=1,\\&p'^{2}+q'^{2}+r'^{2}=1,\\&p''^{2}+q''^{2}+r''^{2}=1,\end{aligned}}\right\}$(5)$\qquad \left.{\begin{aligned}&p'p''+q'q''+r'r''=0,\\&p''p+q''q+r''r=0,\\&pp'+qq'+rr'=0.\end{aligned}}\right\}$(6)

On aura, en substituant,

$\left.{\begin{aligned}&b^{2}c^{2}(\alpha +pt+qu+rv)^{2}\\\\+&c^{2}a^{2}(\beta +p't+q'u+r'v)^{2}\\\\+&a^{2}b^{2}(\gamma +p''t+q''u+r''v)^{2}\end{aligned}}\right\}\quad =a^{2}b^{2}c^{2}.$

En développant les puissances et posant, pour abréger,

$\left.{\begin{aligned}&b^{2}c^{2}p^{2}+c^{2}a^{2}p'^{2}+a^{2}b^{2}p''^{2}=A,\\&b^{2}c^{2}q^{2}+c^{2}a^{2}q'^{2}+a^{2}b^{2}q''^{2}=B,\\&b^{2}c^{2}r^{2}+c^{2}a^{2}r'^{2}+a^{2}b^{2}r''^{2}=C\,;\end{aligned}}\right\}$(7)$\ \left.{\begin{aligned}&b^{2}c^{2}qr+c^{2}a^{2}q'r'+a^{2}b^{2}q''r''=D,\\&b^{2}c^{2}rp+c^{2}a^{2}r'p'+a^{2}b^{2}r''p''=E,\\&b^{2}c^{2}pq+c^{2}a^{2}p'q'+a^{2}b^{2}p''q''=F\,;\end{aligned}}\right\}$(8)

$\left.{\begin{aligned}&b^{2}c^{2}\alpha p+c^{2}a^{2}\beta p'+a^{2}b^{2}\gamma p''=G,\\&b^{2}c^{2}\alpha q+c^{2}a^{2}\beta q'+a^{2}b^{2}\gamma q''=H,\\&b^{2}c^{2}\alpha r+c^{2}a^{2}\beta r'+a^{2}b^{2}\gamma r''=K\,;\end{aligned}}\right\}$(9)

$b^{2}c^{2}\alpha ^{2}+c^{2}a^{2}\beta ^{2}+a^{2}b^{2}\gamma ^{2}-a^{2}b^{2}c^{2}=L\,;\qquad$(10)

on aura