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/75

 == 43. ==

We add, for the better illustration of the preceding investigations, an example of the complete calculation for the true anomaly and radius vector from the time, for which purpose we will resume the numbers in article 38. We put then $e=$ $0.9674567, \log q=9.7656500, t=63.54400,$  whence, we first derive the constants $\log \alpha=0.03052357, \log \beta=8.2217364, \log \gamma=0.0028755.$

Hence we have $\log \alpha t=2.1083102,$ to which corresponds in Barker’s table the approximate value of $v=99^{\circ} 6^{\prime}$  whence is obtained $A=0.022926,$  and from our table $\log B=0.0000040.$  Hence, the correct argument with which Barker’s table must be entered, becomes $\log \frac{\alpha t}{B}=2.1083062,$  to which answers $w=99^{\circ} 6^{\prime}$  $13^{\prime \prime} .14;$  after this, the subsequent calculation is as follows:-

$$\begin{aligned} & \log \tan ^{2} \frac{1}{2} w \quad \text {. . } 0.1385934 \quad \log \tan \frac{1}{2} w \quad \text {. . . . . } 0.0692967 \\ & \log \beta \text {. . . . . } 8.2217364 \quad \log \gamma \text {. . . . . . . } 0.0028755 \\ & \log A \cdot. .8 .3603298 \quad \frac{1}{2} \text { Comp. } \log \left(1-\frac{4}{6} A+C\right) .0 .0040143 \\ & A=\text {. . . . } 0.02292608 \quad \overline{\log \tan \frac{1}{2} v} \text {. . . . . } 0.0761865 \\ & \text { hence } \log B \text { in the same manner as before; } \quad \frac{1}{2} v=\text {. . . . } 50^{\circ} 0^{\prime} 0^{\prime \prime} \\ & C=.0 .0000242 \quad v=. . . . \quad 10000 \\ & 1-\frac{4}{5} A+C=.0 .9816833 \quad \log q \text {. . . . . . } 9.7656500 \\ & 1+\frac{1}{5} A+C=.1 .0046094 \quad \text { 2 Comp. } \log \cos \frac{1}{2} v \text {. . } 0.3838650 \\ & \log \left(1-\frac{4}{6} A+C\right) \text {. . . } 9.9919714 \\ & \text { C. } \log \left(1+\frac{1}{6} A+C\right) \cdot. \quad 9.9980028 \\ & \log r \text {. . . . . . . . } 0.1394892 \end{aligned}$$

If the factor $B$ had been wholly neglected in this calculation, the true anomaly would have come out affected with a very slight error (in excess) of $0^{\prime \prime} .1$  only.

44.
It will be in our power to despatch the hyperbolic motion the more briefly, because it is to be treated in a manner precisely analogous to that which we have thus far expounded for the clliptic motion.