Pagina:Werkecarlf03gausrich.djvu/349

cludi debere valores tales, ubi $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ vel $\beta ,$ $\beta ^{\prime },$ $\beta ^{\prime \prime }$ ipsis $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ resp. proportionales essent: alioquin enim $E$ haud amplius indeterminata maneret. Nequeunt igitur $\gamma ^{\prime }\alpha ^{\prime \prime }-\gamma ^{\prime \prime }\alpha ^{\prime },$ $\gamma ^{\prime \prime }\alpha -\gamma \alpha ^{\prime \prime },$ $\gamma \alpha ^{\prime }-\gamma ^{\prime }\alpha$ simul evanescere.

Manifesto coëfficientes $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ etc. ita comparati esse debent, ut fiat indefinite

$\left.{\begin{array}{r}(\alpha +\alpha ^{\prime }\cos T+\alpha ^{\prime \prime }\sin T)^{2}\\+(\beta +\beta ^{\prime }\cos T+\beta ^{\prime \prime }\sin T)^{2}\\-(\gamma +\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T)^{2}\end{array}}\right\}=0$

unde necessario haec functio habere debet formam

$k({\cos T}^{2}+{\sin T}^{2}-1)$

Hinc colligimus sex aequationes conditionales

$\left.{\begin{alignedat}{5}&-\alpha \alpha &&-\beta \beta &&+\gamma \gamma &&=&k\\&-\alpha ^{\prime }\alpha ^{\prime }&&-\beta ^{\prime }\beta ^{\prime }&&+\gamma ^{\prime }\gamma ^{\prime }&&=&-k\\&-\alpha ^{\prime \prime }\alpha ^{\prime \prime }&&-\beta ^{\prime \prime }\beta ^{\prime \prime }&&+\gamma ^{\prime \prime }\gamma ^{\prime \prime }&&=&-k\\&-\alpha ^{\prime }\alpha ^{\prime \prime }&&-\beta ^{\prime }\beta ^{\prime \prime }&&+\gamma ^{\prime }\gamma ^{\prime \prime }&&=&0\\&-\alpha ^{\prime \prime }\alpha &&-\beta ^{\prime \prime }\beta &&+\gamma ^{\prime \prime }\gamma &&=&0\\&-\alpha \alpha ^{\prime }&&-\beta \beta ^{\prime }&&+\gamma \gamma ^{\prime }&&=&0\end{alignedat}}\right\}$ (I)

Ab his aequationibus pendent plures aliae, quas evolvere operae pretium erit. Statuendo brevitatis caussa

$\alpha \beta ^{\prime }\gamma ^{\prime \prime }+\alpha ^{\prime }\beta ^{\prime \prime }\gamma +\alpha ^{\prime \prime }\beta \gamma ^{\prime }-\alpha \beta ^{\prime \prime }\gamma ^{\prime }-\alpha ^{\prime }\beta \gamma ^{\prime \prime }-\alpha ^{\prime \prime }\beta ^{\prime }\gamma =\varepsilon \ldots \ldots \ldots$ (II)

e combinatione aequationum (I) facile derivantur novem sequentes:

$\left.{\begin{alignedat}{2}&\varepsilon \alpha &&=-k(\beta ^{\prime }\gamma ^{\prime \prime }-\gamma ^{\prime }\beta ^{\prime \prime })\\&\varepsilon \beta &&=-k(\gamma ^{\prime }\alpha ^{\prime \prime }-\alpha ^{\prime }\gamma ^{\prime \prime })\\&\varepsilon \gamma &&=+k(\alpha ^{\prime }\beta ^{\prime \prime }-\beta ^{\prime }\alpha ^{\prime \prime })\\&\varepsilon \alpha ^{\prime }&&=+k(\beta ^{\prime \prime }\gamma -\gamma ^{\prime \prime }\beta )\\&\varepsilon \beta ^{\prime }&&=+k(\gamma ^{\prime \prime }\alpha -\alpha ^{\prime \prime }\gamma )\\&\varepsilon \gamma ^{\prime }&&=-k(\alpha ^{\prime \prime }\beta -\beta ^{\prime \prime }\alpha )\\&\varepsilon \alpha ^{\prime \prime }&&=+k(\beta \gamma ^{\prime }-\gamma \beta ^{\prime })\\&\varepsilon \beta ^{\prime \prime }&&=+k(\gamma \alpha ^{\prime }-\alpha \gamma ^{\prime })\\&\varepsilon \gamma ^{\prime \prime }&&=-k(\alpha \beta ^{\prime }-\beta \alpha ^{\prime })\end{alignedat}}\right\}$ (III)