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/74

 has less effect, in facilitating the calculation, it is not necessary to pay any regard to $A,$ but we have at once

$$\tan \frac{1}{8} w=\tan \frac{1}{3} v \sqrt{\frac{1+C}{\gamma\left(1+\frac{4}{3} T\right)}},$$

and hence the time $t,$ by multiplying the mean notion corresponding to the true anomaly, $w,$  in the Barkerian table, by $\frac{B}{\alpha}.$

42.
We have constructed with sufficient fulness a table, such as we have just described, and have added it to this work, (Table I.). Only the first part pertains to the ellipse; we will explain, further on, the other part, which includes the hyperbolic motion. The argument of the table, which is the quantity $A,$ proceeds by single thousandths from 0 to 0.300 ; the $\log B$  and $C$  follow, which quantities it must be understood are given in ten millionths, or to seven places of decimals, the ciphers preceding the significant figures being suppressed; lastly, the fourth column gives the quantity $T$  computed first to five, then to six figures, which degree of accuracy is quite sufficient, since this column is only needed to get the values of $\log B$  and $C$  corresponding to the argument $T,$  whenever $t$  is to be determined from $v$  by the precept of the preceding article. As the inverse problem which is much more frequently employed, that is, the determination of $v$ and $r$  from $t,$  is solved altogether without the help of $T,$  we have preferred the quantity $A$  for the argument of our table rather than $T,$  which would otherwise have been an almost equally suitable argument, and would even have facilitated a little the construction of the table. It will not be unnecessary to mention, that all the numbers of the table have been calculated from the beginning to ten places, and that, therefore, the seven places of figures which we give can be safely relied upon; but we cannot dwell here upon the analytical methods used for this work, by a full explanation of which we should be too much diverted from our plan. Finally, the extent of the table is abundantly sufficient for all cases in which it is advantageous to pursue the method just explained, since beyond the limit $A=0.3,$ to which answers $T=0.392374,$  or $E=64^{\circ} 7^{\prime},$  we may, as has been shown before, conveniently dispense with artificial methods.