Pagina:Werkecarlf03gausrich.djvu/353

in

$G+G^{\prime }{\cos T}^{2}+G^{\prime \prime }{\sin T}^{2}$

Manifesto hoc idem est, ac si dicas, functionem trium indeterminatarum $x,$ $y,$ $z$ hanc $(W)$

$aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz$

per substitutionem

${\begin{aligned}&x=\alpha u+\alpha ^{\prime }u^{\prime }+\alpha ^{\prime \prime }u^{\prime \prime }\\&y=\beta u+\beta ^{\prime }u^{\prime }+\beta ^{\prime \prime }u^{\prime \prime }\\&z=\gamma u+\gamma ^{\prime }u^{\prime }+\gamma ^{\prime \prime }u^{\prime \prime }\end{aligned}}$

in functionem indeterminatarum $u,$ $u^{\prime },$ $u^{\prime \prime }$ hanc

$Guu+G^{\prime }u^{\prime }u^{\prime }+G^{\prime \prime }u^{\prime \prime }u^{\prime \prime }$

transire debere. At quum ex his formulis, adiumento aequationum [1], facile sequatur

${\begin{alignedat}{8}&u&&=-&&\alpha x-\beta y+\gamma z\\&u^{\prime }&&=&&\alpha ^{\prime }x+\beta ^{\prime }y-\gamma ^{\prime }z\\&u^{\prime \prime }&&=&&\alpha ^{\prime \prime }x+\beta ^{\prime \prime }y-\gamma ^{\prime \prime }z\end{alignedat}}$

manifesto functio $W$ identica esse debebit cum hac

$G(-\alpha x-\beta y+\gamma z)^{2}+G^{\prime }(\alpha ^{\prime }x+\beta ^{\prime }y-\gamma ^{\prime }z)^{2}+G^{\prime \prime }(\alpha ^{\prime \prime }x+\beta ^{\prime \prime }y-\gamma ^{\prime \prime }z)^{2}$

unde habemus sex aequationes

$\left.{\begin{array}{rl}aa&=G\alpha \alpha +G^{\prime }\alpha ^{\prime }\alpha ^{\prime }+G^{\prime \prime }\alpha ^{\prime \prime }\alpha ^{\prime \prime }\\bb&=G\beta \beta +G^{\prime }\beta ^{\prime }\beta ^{\prime }+G^{\prime \prime }\beta ^{\prime \prime }\beta ^{\prime \prime }\\AA+BB+CC&=G\gamma \gamma +G^{\prime }\gamma ^{\prime }\gamma ^{\prime }+G^{\prime \prime }\gamma ^{\prime \prime }\gamma ^{\prime \prime }\\bB&=G\beta \gamma +G^{\prime }\beta ^{\prime }\gamma ^{\prime }+G^{\prime \prime }\beta ^{\prime \prime }\gamma ^{\prime \prime }\\aA&=G\gamma \alpha +G^{\prime }\gamma ^{\prime }\alpha ^{\prime }+G^{\prime \prime }\gamma ^{\prime \prime }\alpha ^{\prime \prime }\\0&=G\alpha \beta +G^{\prime }\alpha ^{\prime }\beta ^{\prime }+G^{\prime \prime }\alpha ^{\prime \prime }\beta ^{\prime \prime }\end{array}}\right\}$ [2]

Ex his duodecim aequationibus [1] et [2] incognitas nostras $G,$ $G^{\prime },$ $G^{\prime \prime },$ $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ etc., determinare oportebit.