Pagina:Werkecarlf03gausrich.djvu/362

$aa-G^{\prime },bb-G^{\prime },aa-G^{\prime \prime },bb-G^{\prime \prime }$ evanescit, valor coëfficientis $\alpha ^{\prime },\beta ^{\prime },\alpha ^{\prime \prime },\gamma ^{\prime \prime }$ per formulam $6,7,9,10$ resp. indeterminatus manere primo aspectu videtur, quod tamen secus se habere levis attentio docebit.

Supponumus e.g., esse $aa-G^{\prime }=0,$ fietque, per aequationem $18,\gamma ^{\prime }=0,$ nec non per aequationem 7, $\beta ^{\prime }=0$ (siquidem non fuerit simul $aa=bb$) unde necessario esse debet $\alpha ^{\prime }=\pm 1.$ Si vero simul $aa=bb,$ formula, quae praecedit sextam in art. 5, suppeditat $\alpha ^{\prime }A+\beta ^{\prime }B=0,$ quae aequatio cum $\alpha ^{\prime }\alpha ^{\prime }+\beta ^{\prime }\beta ^{\prime }=1$ iuncta, producit

$\alpha ^{\prime }={\frac {B}{\sqrt {AA+BB}}},\quad \beta ^{\prime }={\frac {-A}{\sqrt {AA+BB}}}$

Hae expressiones manifesto indeterminatae esse nequeunt, nisi simul fuerit $A=0,$ $B=0;$ tunc vero ad casum in art. praec. iam consideratum delaberemur.

12.

Postquam duodecim quantitates $G,$ $G^{\prime },$ $G^{\prime \prime },$ $\alpha ,$ $\alpha ^{\prime },$ $\alpha ^{\prime \prime },$ $\beta ,$ $\beta ^{\prime },$ $\beta ^{\prime \prime },$ $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ complete determinare docuimus, ad evolutionem differentialis $\operatorname {d} E$ progredimur. Statuamus

$t=\gamma +\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T\ldots \ldots \ldots \ldots$ [20]

ita ut fiat

$t\cos E=\alpha +\alpha ^{\prime }\cos T+\alpha ^{\prime \prime }\sin T\ldots \ldots$	[21]
$t\sin E=\beta +\beta ^{\prime }\cos T+\beta ^{\prime \prime }\sin T\,\ldots \ldots$	[22]

Hinc deducimus

${\begin{aligned}t\operatorname {d} E&=\cos E\operatorname {d} .t\sin E-\sin E\operatorname {d} .t\cos E\\&=\cos E(\beta ^{\prime \prime }\cos T-\beta ^{\prime }\sin T)\operatorname {d} T-\sin E(\alpha ^{\prime \prime }\cos T-\alpha ^{\prime }\sin T)\operatorname {d} T\end{aligned}}$

adeoque

${\begin{aligned}tt\operatorname {d} E&=(\alpha \beta ^{\prime \prime }-\alpha ^{\prime \prime }\beta )\cos T\operatorname {d} T+(\alpha ^{\prime }\beta -\beta ^{\prime }\alpha )\sin T\operatorname {d} T+(\alpha ^{\prime }\beta ^{\prime \prime }-\beta ^{\prime }\alpha ^{\prime \prime })\operatorname {d} T\\&=\varepsilon \gamma ^{\prime }\cos T\operatorname {d} T+\varepsilon \gamma ^{\prime \prime }\sin T\operatorname {d} T+\varepsilon \gamma \operatorname {d} T=\varepsilon t\operatorname {d} T\end{aligned}}$

sive

$t\operatorname {d} E=\varepsilon \operatorname {d} T\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots [23]$

Observare convenit, quantitatem $t$ natura sua semper positivam esse, si coëfficiens $\gamma$ sit positivus, vel semper negativam, si $\gamma$ sit negativus. Quum enim sit

$(\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T)^{2}+(\gamma ^{\prime \prime }\cos T-\gamma ^{\prime }\sin T)^{2}=\gamma ^{\prime }\gamma ^{\prime }+\gamma ^{\prime \prime }\gamma ^{\prime \prime }=\gamma \gamma -1,$

erit semper