Pagina:Werkecarlf03gausrich.djvu/351

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

Combinatio reliquarum easdem produceret. His denique adiungimus tres sequentes:

$\left.{\begin{aligned}\beta \gamma -\beta ^{\prime }\gamma ^{\prime }-\beta ^{\prime \prime }\gamma ^{\prime \prime }&=0\\\gamma \alpha -\gamma ^{\prime }\alpha ^{\prime }-\gamma ^{\prime \prime }\alpha ^{\prime \prime }&=0\\\alpha \beta -\alpha ^{\prime }\beta ^{\prime }-\alpha ^{\prime \prime }\beta ^{\prime \prime }&=0\end{aligned}}\right\}$ (VII)

quae facile ex aequationibus III derivantur; e.g. secunda, quinta et octava suppeditant:

$\varepsilon \beta \gamma -\varepsilon \beta ^{\prime }\gamma ^{\prime }-\varepsilon \beta ^{\prime \prime }\gamma ^{\prime \prime }=-k\gamma (\gamma ^{\prime }\alpha ^{\prime \prime }-\alpha ^{\prime }\gamma ^{\prime \prime })-k\gamma ^{\prime }(\gamma ^{\prime \prime }\alpha -\alpha ^{\prime \prime }\gamma )-k\gamma ^{\prime \prime }(\gamma \alpha ^{\prime }-\alpha \gamma ^{\prime })=0$

Manifesto hae quoque aequationes ab exclusione valoris $k=0$ sunt dependentes^[1]

Quoniam, ut iam supra monuimus, omnes coëfficientes $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ etc. per eundem factorem multiplicare licet, unde valor ipsius $k$ per quadratum eiusdem factoris multiplicatus prodibit, abhinc semper supponemus

$k=1$

quo pacto necessario quoque erit vel $\varepsilon =+1$ vel $\varepsilon =-1.$ Patet itaque, novem coëfficientes $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ etc., inter quos sex aequationes conditionales adsunt, ad tres quantitates ab invicem independentes reducibiles esse debere, quod quidem commodissime per tres angulos sequenti modo efficitur:

${\begin{aligned}\alpha &=\cos L\operatorname {tang} N\\\beta &=\sin L\operatorname {tang} N\\\gamma &=\sec N\\\alpha ^{\prime }&=\cos L\cos M\sec N\pm \sin L\sin M\\\beta ^{\prime }&=\sin L\cos M\sec N\mp \cos L\sin M\\\gamma ^{\prime }&=\cos M\operatorname {tang} N\\\alpha ^{\prime \prime }&=\cos L\sin M\sec N\mp \sin L\cos M\\\beta ^{\prime \prime }&=\sin L\sin M\sec N\pm \cos L\cos M\\\gamma ^{\prime \prime }&=\sin M\operatorname {tang} N\end{aligned}}$

1. Forsan haud superfluum erit monere, nos analysin praecedentem consulto elegisse atque alii derivationi relationum III-VII praetulisse, quae quamquam aliquantulum elegantior videretur, tamen, accurate examinata, quibusdam dubiis obnoxia inventa est, quae non sine ambagibus removere licuisset.