Pagina:Werkecarlf03gausrich.djvu/359

Postquam coëfficientes $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ inventi sunt, reliqui $\alpha ,$ $\beta ,$ $\alpha ^{\prime },$ $\beta ^{\prime },$ $\alpha ^{\prime \prime },$ $\beta ^{\prime \prime }$ inde per formulas 3, 4, 6, 7, 9, 10 derivabuntur.

9.

Signa expressionum radicalium, per quas $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ determinavimus, ad lubitum accipi posse facile perspicitur. Operae autem pretium est, inquirere, quomodo signum quantitatis $\varepsilon$ cum signis istis nexum sit. Ad hunc finem consideremus aequationem tertiam in III art. 3.

$\varepsilon \gamma =\alpha ^{\prime }\beta ^{\prime \prime }-\beta ^{\prime }\alpha ^{\prime \prime }$

quae per formulas 6, 7, 9, 10 transmutatur in hanc:

${\begin{aligned}\varepsilon \gamma &={\frac {abAB\gamma ^{\prime }\gamma ^{\prime \prime }}{(aa-G^{\prime })(bb-G^{\prime \prime })}}-{\frac {abAB\gamma ^{\prime }\gamma ^{\prime \prime }}{(aa-G^{\prime \prime })(bb-G^{\prime })}}\\&={\frac {ab(aa-bb)AB(G^{\prime \prime }-G^{\prime })\gamma ^{\prime }\gamma ^{\prime \prime }}{(aa-G^{\prime })(aa-G^{\prime \prime })(bb-G^{\prime })(bb-G^{\prime \prime })}}\end{aligned}}$

Sed e consideratione aequationis 13 facile deducimus

${\begin{aligned}&(aa+G)(aa-G^{\prime })(aa-G^{\prime \prime })=aa(aa-bb)AA\\&(bb+G)(bb-G^{\prime })(bb-G^{\prime \prime })=-bb(aa-bb)BB\end{aligned}}$

Hinc aequatio praecedens fit

$\varepsilon \gamma ={\frac {(aa+G)(bb+G)(G^{\prime }-G^{\prime \prime })\gamma ^{\prime }\gamma ^{\prime \prime }}{ab(aa-bb)AB}}$

quae combinata cum aequatione 17 suppeditat

$\gamma \gamma ^{\prime }\gamma ^{\prime \prime }={\frac {\varepsilon ab(aa-bb)AB}{(G+G^{\prime })(G+G^{\prime \prime })(G^{\prime }-G^{\prime \prime })}}$

Hinc patet, si pro $-G^{\prime }$ electa sit aequationis cubicae radix negativa absolute maior, simulque coëfficientes $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ omnes positive accepti sint, $\varepsilon$ idem signum nancisci, quod habet $AB,$ idemque evenire, si his quatuor conditionibus, vel omnibus vel duabus ex ipsis, contraria acta sint, oppositum vero, si uni vel tribus conditionibus adversatus fueris. Ceterum sequentes adhuc relationes notare convenit, e praecedentibus facile derivandas: