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/42

 the former when $\sin Q$ is greater than $\cos Q;$  the latter when $\cos Q$  is greater than $\sin Q.$  Commonly, the problems in which equations of this kind occur (such as present themselves most frequently in this work), involve the condition that $P$  should be a positive quantity; in this case, the doubt whether $Q$  should be taken between 0 and $180^{\circ},$  or between $180^{\circ}$  and $360^{\circ},$  is at once removed. But if such a condition does not éxist, this decision is left to our judgment.

We have in our example $e=0.2453162.$

$$\begin{aligned} & \log \sin \frac{1}{2} E && 9.486732 && \log \cos \frac{1}{2} E && 9.486732 n \\ & \log \sqrt{a \left ( 1+e \right ) } && 0.2588593 && \log \sqrt{a \left ( 1+e \right ) } &&  0.1501020 \\ \end{aligned} $$

Hence $$\begin{aligned}

& \log \sin \frac{1}{2} v\sqrt{r} && 9.7456225 && \text {whence} && \log \tan\frac{1}{2} v = 9.169771 n \\ & \ log \cos \frac{1}{2} v\sqrt{r} && 0.1286454 n && && \frac{1}{2}v = 157^\circ 30' 41''.50 \\ & \ log \cos \frac{1}{2} v && 9.9656515 n && && v = 315^\circ 1' 23''.00 \\ & \log \sqrt{r} && 0.1629939 \\ & \log r && 0.3259878 \\ \end{aligned} $$

III. To these methods we add a third which is almost equally easy and expeditious, and is much to be preferred to the former if the greatest accuracy should be required. Thus, $r$ is first determined by means of equation III, and after that, $v$  by $\mathrm{X}.$  Below is our example treated in this manner.



Formula VIII, or XI, is very convenient for verifying the calculation, particularly if $v$ and $r$  have been determined by the third method. Thus;