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/52

 Example. - Let $e=1.2618820,$ or $\psi=37^{\circ} 35^{\prime} 0^{\prime \prime}, v=18^{\circ} 51^{\prime} 0^{\prime \prime}, \log r=$  0.0333585. Then the computation for $u, p, b, N, t,$ is as follows:-



First term of $N=.$ 0.0621069

$\log u=\quad 0.0491129$

$$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \multirow{4}{*}{}\begin{array}{l} N= \\ \log \lambda k \\ \frac{3}{2} \log b \\ \end{array} & & \therefore & 0.0129940 & \log N &. & & . & 8.1137429 \\ \hline & . . & - . & 7.8733658\} & &  &  &  &  \\ \hline & \cdot &. . & 0.9030900\} & Difference & \includegraphics[max width=\textwidth]{2024_02_19_0baa176114aa2645ceebg-048} & - & & 6.9702758 \\ \hline & &  &  & \begin{array}{l} \log t \\ t= \\ \end{array} & &  &  & \begin{array}{r} 1.1434671 \\ 13.91448 \\ \end{array} \\ \hline \end{array}$$

24.
If it has been decided to carry out the calculation with hyperbolic logarithms, it is best to employ the auxiliary quantity $F,$ which will be determined by equation III., and thence $N$  by XI.; the semi-parameter will be computed from the radius vector, or inversely the latter from the former by formula VIII.; the second part of $N$  can, if desired, be obtained in two ways, namely, by means of the formula hyp. $\log \tan \left(45^{\circ}+\frac{1}{2} F\right),$ and by this, hyp. $\log \cos \frac{1}{2}(v-\psi)-$ hyp. $\log$ $\cos \frac{1}{2}(v+\psi).$  Moreover it is apparent that here where $\lambda=1$  the quantity $N$