Pagina:Werkecarlf03gausrich.djvu/364

14.

E combinatione aequationum 20, 21, 22 cum aequationibus art. 5 obtinemus:

${\begin{aligned}&at(A-a\cos E)=\alpha G-\alpha ^{\prime }G^{\prime }\cos T-\alpha ^{\prime \prime }G^{\prime \prime }\sin T\\&bt(B-b\sin E)=\beta G-\beta ^{\prime }G^{\prime }\cos T-\beta ^{\prime \prime }G^{\prime \prime }\sin T\end{aligned}}$

Statuendo itaque brevitatis gratia

${\begin{aligned}(\alpha G-\alpha ^{\prime }G^{\prime }\cos T-\alpha ^{\prime \prime }G^{\prime \prime }\sin T)(\gamma -e\alpha +(\gamma ^{\prime }-e\alpha ^{\prime })\cos T+(\gamma ^{\prime \prime }-e\alpha ^{\prime \prime })\sin T)&=aX\\(\beta G-\beta ^{\prime }G^{\prime }\cos T-\beta ^{\prime \prime }G^{\prime \prime }\sin T)(\gamma -e\alpha +(\gamma ^{\prime }-e\alpha ^{\prime })\cos T+(\gamma ^{\prime \prime }-e\alpha ^{\prime \prime })\sin T)&=bY\\C(\gamma +\gamma ^{\prime }\cos T+\gamma ^{\prime \prime }\sin T)(\gamma -e\alpha +(\gamma ^{\prime }-e\alpha ^{\prime })\cos T+(\gamma ^{\prime \prime }-e\alpha ^{\prime \prime })\sin T)&=Z\end{aligned}}$

fit

$\operatorname {d} \xi ={\frac {\varepsilon X\operatorname {d} T}{2\pi t^{3}\rho ^{3}}},\quad$$\operatorname {d} \eta ={\frac {\varepsilon Y\operatorname {d} T}{2\pi t^{3}\rho ^{3}}},\quad$$\operatorname {d} \zeta ={\frac {\varepsilon Z\operatorname {d} T}{2\pi t^{3}\rho ^{3}}}$

Sed habetur

$t\rho =\pm {\sqrt {G+G^{\prime }\cos T^{2}+G^{\prime \prime }\sin T^{2}}}$

signo superiore vel inferiore valente, prout $t$ est quantitas positiva vel negativa ($\rho$ enim natura sua semper positive accipitur), i.e. prout coëfficiens $\gamma$ est positivus vel negativus. Hinc

${\frac {\varepsilon \operatorname {d} T}{2\pi t^{3}\rho ^{3}}}=\pm {\frac {\operatorname {d} T}{2\pi (G+G^{\prime }\cos T^{2}+G^{\prime \prime }\sin T^{2})^{\frac {3}{2}}}}$

ubi signum ambiguum a signo quantitatis $\gamma \varepsilon$ pendet.

Ut iam valores ipsarum $\xi ,$ $\eta ,$ $\zeta$ obtineamus, integrationes differentialium exsequi oportet, a valore ipsius $T,$ cui respondet $E=0,$ usque ad valorem, cui respondet $E=360^{\circ },$ sive etiam (quod manifesto eodem redit) a valore ipsius $T,$ cui respondet valor arbitrarius ipsius $E,$ usque ad valorem, cui respondet valor ipsius $E$ auctus $360^{\circ };$ licebit itaque integrare a $T=0$ usque ad $T=360^{\circ },$ quoties $\varepsilon \gamma$ est quantitas positiva, vel a $T=360^{\circ }$ usque ad $T=0,$ quoties $\varepsilon \gamma$ est negativa. Manifesto itaque, independenter a signo ipsius $\varepsilon \gamma ,$ erit:

${\begin{aligned}&\xi =\int {\frac {X\operatorname {d} T}{2\pi (G+G^{\prime }\cos T^{2}+G^{\prime \prime }\sin T^{2})^{\frac {3}{2}}}}\\&\eta =\int {\frac {Y\operatorname {d} T}{2\pi (G+G^{\prime }\cos T^{2}+G^{\prime \prime }\sin T^{2})^{\frac {3}{2}}}}\\&\zeta =\int {\frac {Z\operatorname {d} T}{2\pi (G+G^{\prime }\cos T^{2}+G^{\prime \prime }\sin T^{2})^{\frac {3}{2}}}}\end{aligned}}$

integrationibus a $T=0$ usque ad $T=360^{\circ }$ extensis.