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/77

 45.

The latter part of the table annexed to this work belongs, as we have remarked above, to the hyperbolic motion, and gives for the argument $A$ (common to both parts of the table), the logarithm of $B$  and the quantity $C$  to seven places of decimals, (the preceding ciphers being omitted), and the quantity $T$  to five and afterwards to six figures. The latter part is extended in the same manner as the former to $A=0.300,$ corresponding to which is $T=0.241207, u=2.930,$  or $=0.341, F= \pm 52^{\circ} 19^{\prime};$  to extend it further would have been superfluous, (article 36 ).

The following is the arrangement of the calculation, not only for the determination of the time from the true anomaly, but for the determination of the true anomaly from the time. In the former problem, $T$ will be got by means of the formula

$$T=\frac{e-1}{e+1} \tan ^{2} \frac{1}{2} v$$

with $T$ our table will give $\log B$  and $C,$  whence will follow

$$A=\frac{(1+C) T}{1-\frac{C}{5} T} ;$$

finally $t$ is then found from the formula [2] of the preceding article. In the last problem, will first be computed, the logarithms of the constants

$$\begin{aligned} & \alpha=\frac{75 k \sqrt{ }\left(\frac{1}{5}+\frac{9}{5} e\right)}{2 q^{\frac{3}{2}}} \\ & \beta=\frac{5 e-5}{\mathrm{I}+9 e} \\ & \gamma=\sqrt{\frac{5 e+5}{1+9 e}} \end{aligned}$$

$A$ will then be determined from $t$  exactly in the same manner as in the elliptic motion, so that in fact the true anomaly $w$  may correspond in Barker’s table to the mean motion $\frac{\alpha t}{B},$  and that we may have

$$A=\beta \tan ^{2} \frac{1}{2} w ;$$

the approximate value of $A$ will be of course first obtained, the factor $B$  being