Pagina:Werkecarlf03gausrich.djvu/350

E tribus primis harum aequationum rursus deducimus hanc:

${\begin{aligned}\varepsilon \alpha (\beta ^{\prime }\gamma ^{\prime \prime }-\gamma ^{\prime }\beta ^{\prime \prime })+\varepsilon \beta (\gamma ^{\prime }\alpha ^{\prime \prime }-\alpha ^{\prime }\gamma ^{\prime \prime })+\varepsilon \gamma (\alpha ^{\prime }\beta ^{\prime \prime }-\beta ^{\prime }\alpha ^{\prime \prime })\\=-k(\beta ^{\prime }\gamma ^{\prime \prime }-\gamma ^{\prime }\beta ^{\prime \prime })^{2}-k(\gamma ^{\prime }\alpha ^{\prime \prime }-\alpha ^{\prime }\gamma ^{\prime \prime })^{2}+k(\alpha ^{\prime }\beta ^{\prime \prime }-\beta ^{\prime }\alpha ^{\prime \prime })^{2}\end{aligned}}$

cui aequivalens est haee:

$\varepsilon \varepsilon =k(-\alpha ^{\prime }\alpha ^{\prime }-\beta ^{\prime }\beta ^{\prime }+\gamma ^{\prime }\gamma ^{\prime })(-\alpha ^{\prime \prime }\alpha ^{\prime \prime }-\beta ^{\prime \prime }\beta ^{\prime \prime }+\gamma ^{\prime \prime }\gamma ^{\prime \prime })-k(-\alpha ^{\prime }\alpha ^{\prime \prime }-\beta ^{\prime }\beta ^{\prime \prime }+\gamma ^{\prime }\gamma ^{\prime \prime })^{2}$

quae adiumento aequationum 2, 3, 4 in (I) mutatur in hanc:

$\varepsilon \varepsilon =k^{3}\ldots \ldots \ldots \ldots$ (IV)

Aeque facile ex aequationibus (I) derivantur hae:

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

Exempli caussa evolutionem primae adscribimus, ad cuius instar reliquae facile formabuntur. Aequationes 4, 2, 3 in (I) scilicet suppeditant

$(\gamma ^{\prime }\gamma ^{\prime \prime }-\beta ^{\prime }\beta ^{\prime \prime })^{2}-(\gamma ^{\prime }\gamma ^{\prime }-\beta ^{\prime }\beta ^{\prime })(\gamma ^{\prime \prime }\gamma ^{\prime \prime }-\beta ^{\prime \prime }\beta ^{\prime \prime })=\alpha ^{\prime }\alpha ^{\prime }\alpha ^{\prime \prime }\alpha ^{\prime \prime }-(\alpha ^{\prime }\alpha ^{\prime }-k)(\alpha ^{\prime \prime }\alpha ^{\prime \prime }-k)$

quae aequatio evoluta protinus ipsam primam in (V) sistit.

Ex his aequationibus (V) concludimus, valorem $k=0$ in disquisitione nostra haud admissibilem esse; hinc enim omnes novem quantitates $\beta ^{\prime }\gamma ^{\prime \prime }-\gamma ^{\prime }\beta ^{\prime \prime }$ etc. necessario evanescerent, i.e. coëfficientes $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ tum ipsis $\beta ,$ $\beta ^{\prime },$ $\beta ^{\prime \prime },$ tum ipsis $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ proportionales evaderent. Hinc etiam, propter aequationem IV, quantitas $\varepsilon$ evanescere nequit; quamobrem $k$ necessario debet esse quantitas positiva, siquidem omnes coëfficientes $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ etc. debent esse reales. Combinatis tribus aequationibus primis in (III) cum tribus primis in (V), hae novae prodeunt, quae manifesto a valore ipsius $k$ non evanescente pendent: