Pagina:Werkecarlf03gausrich.djvu/352

ubi signorum ambiguorum superiora referuntur ad casum $\varepsilon =+1,$ inferiora ad casum $\varepsilon =-1.$ Attamen tractatio analytica ad maximam partem elegantius sine usu horum angulorum absolvitur. Ceterum haud difficile foret, significationem geometricam tum horum angulorum, tum reliquarum quantitatum auxiliarium in hac disquisitione occurrentium assignare; hanc vero interpretationem ad institutum nostrum haud necessariam lectori perito explicandam linquimus.

4.

Si iam in expressione distantiae $\rho$ pro $\cos E$ et $\sin E$ valores supra assumti substituuntur, illa in hanc formam transibit:

$\rho ={\frac {\sqrt {G+G^{\prime }\cos T^{2}+G^{\prime \prime }\sin T^{2}+2H\cos T\sin T+2H^{\prime }\sin T+2H^{\prime \prime }\cos T}}{\gamma +\gamma ^{\prime }\cos T^{\prime }+\gamma ^{\prime \prime }\sin T}}$

ubi coëfficientes $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime }$ etc. ita determinabimus, ut salvis sex aequationibus conditionalibus

$\left.{\begin{alignedat}{7}&-\alpha \alpha &&-\beta \beta &&+\gamma \gamma &&=&1\\&-\alpha ^{\prime }\alpha ^{\prime }&&-\beta ^{\prime }\beta ^{\prime }&&+\gamma ^{\prime }\gamma ^{\prime }&&=&-1\\&-\alpha ^{\prime \prime }\alpha ^{\prime \prime }&&-\beta ^{\prime \prime }\beta ^{\prime \prime }&&+\gamma ^{\prime \prime }\gamma ^{\prime \prime }&&=&\,-1\\&-\alpha ^{\prime }\alpha ^{\prime \prime }&&-\beta ^{\prime }\beta ^{\prime \prime }&&+\gamma ^{\prime }\gamma ^{\prime \prime }&&=&0\\&-\alpha ^{\prime \prime }\alpha &&-\beta ^{\prime \prime }\beta &&+\gamma ^{\prime \prime }\gamma &&=&0\\&-\alpha \alpha ^{\prime }&&-\beta \beta ^{\prime }&&+\gamma \gamma ^{\prime }&&=&0\end{alignedat}}\right\}$ [1]

adeoque etiam reliquis inde demanantibus, fiat

$H=0,\quad H^{\prime }=0,\quad H^{\prime \prime }=0$

quo pacto problema generaliter loquendo erit determinatum. Quodsi itaque denominatorem ipsius $\rho$ per $t$ denotamus, transire debet functio trium quantitatum $t,$ $t\cos E,$ $t\sin E$ haec

$(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt.t\cos E-2bBt.t\sin E$

per substitutionem

${\begin{alignedat}{2}&t\cos E&&=\alpha +\alpha ^{\prime }\cos T+\alpha ^{\prime \prime }\sin T\\&t\sin E&&=\beta +\beta ^{\prime }\cos T+\beta ^{\prime \prime }\sin T\\&t&&=\gamma +\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T\end{alignedat}}$