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

image

First term of ${\textstyle N=.}$ 0.0621069

${\textstyle \log u=\quad 0.0491129}$

Failed to parse (unknown function "\multirow"): {\displaystyle \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 ${\textstyle F,}$ which will be determined by equation III., and thence ${\textstyle N}$ 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 ${\textstyle N}$ can, if desired, be obtained in two ways, namely, by means of the formula hyp. ${\textstyle \log \tan \left(45^{\circ }+{\frac {1}{2}}F\right),}$ and by this, hyp. ${\textstyle \log \cos {\frac {1}{2}}(v-\psi )-}$ hyp. ${\textstyle \log }$ ${\textstyle \cos {\frac {1}{2}}(v+\psi ).}$ Moreover it is apparent that here where ${\textstyle \lambda =1}$ the quantity ${\textstyle N}$