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

We have in our example ${\textstyle e=0.2453162.}$

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

Hence

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

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, ${\textstyle r}$ is first determined by means of equation III, and after that, ${\textstyle v}$ by ${\textstyle \mathrm {X} .}$ Below is our example treated in this manner.

image

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