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/78

 either neglected, or, if the means are at hand, being estimated; our table will then furnish the approximate value of $B,$ with which the work will be repeated; the new value of $B$  resulting in this manner will scarcely ever suffer sensible correction, and thus a second repetition of the calculation will not be necessary. $C$ will be taken from the table with the corrected value of $A,$  which being done we shall have,

$$\tan \frac{1}{2} v=\frac{\gamma \tan \frac{1}{2} w}{\sqrt{\left(1+\frac{4}{3} A+O\right)}}, r=\frac{\left(1+\frac{4}{5} A+C\right) q}{\left(1-\frac{1}{8} A+C\right) \cos ^{2} \frac{1}{2} v}$$

From this it is evident, that no difference can be perceived between the formulas for elliptic and hyperbolic motions, provided that we consider $\beta, A,$ and $T,$  in the hyperbolic motion as negative quantities.

46.
It will not be unprofitable to elucidate the hyperbolic motion also by some examples, for which purpose we will resume the numbers in articles $23,26.$

I. The data are $e=1.2618820, \log q=0.0201657, v=18^{\circ} 51^{\prime} 0^{\prime \prime}: t$ is required. We have

$$\begin{aligned} & 2 \log \tan \frac{1}{2} v \quad \text {. . . . } 8.4402018 \quad && \log T \text {. . . . . } 7.5038375 \\ & \log \frac{e-1}{e+1} \text {. . . . . . } 9.0636357 \quad && \log (1+C) \text {. . } 0.0000002 \\ & \log T \text {. . . . . }7.503875 && \mathrm{C}. \log \left (1 - \frac{4}{5} T \right ) \\ & \log B=\text {. . . . . } 0.0000001 && \log A \text {. . . . . } 7.5049476 \\ & C=\text {. . . . } 0.0000005 \\ & \log \frac{2 B q^{\frac{3}{2}}}{k \sqrt{ }(e-1)} \cdot .2 .3866444 \log \frac{2 B(1+9 e)}{15 k}\left(\frac{q}{e-1}\right)^{\frac{9}{2}} \cdot. \cdot 2.8843582 \\ & \log A^{\frac{1}{2}} \text {. . . . . } 8.7524738 . \log A^{\frac{3}{2}} \text {. . . . . . . . } 6.2574214 \\ & \log 13.77584=. .1 .1391182 \log 0.138605=. . . . .9 .1417796 \\ & 0.13861 \\ & 13.91445=t. \end{aligned}$$

II. $e$ and $q$  remaining as before, there is given $t=65.41236 ; v$  and $r$  are required. We find the logarithms of the constants,