Page:Theory of the motion of the heavenly bodies moving about the sun in conic sections- a translation of Gauss's "Theoria motus." With an appendix (IA theoryofmotionof00gaus).pdf/51

the mass ${\textstyle \mu }$ (which we can assume to be indeterminable for a body moving in an hyperbola) is neglected, the equation assumes the following form:-

XI. ${\textstyle {\frac {1}{2}}\lambda e{\frac {uu-1}{u}}-\log u={\frac {\lambda kt}{b^{\frac {3}{2}}}},}$

or by introducing ${\textstyle F,}$

$$\lambda e\tan F-\log \tan \left(45^{\circ }+{\frac {1}{3}}F^{\prime }\right)={\frac {\lambda kt}{b^{\frac {3}{2}}}}{\text{. }}$$

Supposing Brigg’s logarithms to be used, we have

$$\log \lambda =9.6377843113,\quad \log \lambda k=7.8733657527$$

but a little greater precision can be attained by the immediate application of the

• hyperbolic logarithms. The hyperbolic logarithms of the tangents are found in several collections of tables, in those, for example, which Schutze edited, and still more extensively in the Magnus Canon Triangulor. Iogarithmicus of Benjamin Ursin, Cologne, 1624, in which they proceed by tens of seconds.

Finally, formula XI. shows that opposite values of ${\textstyle t}$ correspond to reciprocal values of ${\textstyle u,}$ or opposite values of ${\textstyle F}$ and ${\textstyle v,}$ on which account equal parts of the hyperbola, at equal distances from the perihelion on both sides, are described in equal times.

23.

If we should wish to make use of the auxiliary quantity ${\textstyle u}$ for finding the time from the true anomaly, its value is most conveniently determined by means of equation IV.; afterwards, formula II. gives directly, without a new calculation, ${\textstyle p}$ by means of ${\textstyle r,}$ or ${\textstyle r}$ by means of ${\textstyle p.}$ Having found ${\textstyle u,}$ formula XI. will give the quantity ${\textstyle {\frac {\lambda kt}{b^{\frac {3}{2}}}},}$ which is analogous to the mean anomaly in the ellipse and will be denoted by ${\textstyle N,}$ from which will follow the elapsed time after the perihelion transit. Since the first term of ${\textstyle N,}$ that is ${\textstyle {\frac {2e(uu-1)}{2u}}}$ may, by means of formula VIII. be made ${\textstyle ={\frac {\lambda r\sin v}{b\sin \psi }},}$ the double computation of this quantity will answer for testing its accuracy, or, if preferred, ${\textstyle N}$ can be expressed without ${\textstyle u,}$ as follows: -

XII. ${\textstyle N={\frac {\lambda \cdot \tan \psi \sin v}{2\cos {\frac {1}{2}}(v+\psi )\cos {\frac {1}{2}}(v-\psi )}}-\log {\frac {\cos {\frac {1}{2}}(v-\psi )}{\cos {\frac {1}{2}}(v+\psi )}}.}$