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/72

 auxiliary table, with which we will repeat more exactly the same calculation; most frequently; precisely the same value of $B$ that had been found from the approximate value of $A$  will correspond to the value of $A$  thus corrected, so that a second repetition of the operation would be superfluous, those cases excepted in which the value of $E$  may have been very considerable.

Finally, it is hardly necessary to observe that, if the approximate value of $B$ should in any other way whatever be known from the beginning, (which may always occur, when of several places to be computed, not very distant from each other, some few are already obtained,) it is better to make use of this at once in the first approximation: in this manner the expert computer will very often not have occasion for even a single repetition. We have arrived at this most rapid approximation from the fact that $B$ differs from unity, only by a difference of the fourth order, and is multiplied by a very small numerical coefficient, which advantage, as will now be perceived, was secured by the introduction of the quantities $E-\sin E, \frac{9}{10} E+\frac{1}{10} \sin E,$  in the place of $E$  and $\sin E.$

40.
Since, for the third operation, that is, the determination of the true anomaly, the angle $E$ is not required, but the $\tan \frac{1}{2} E$  only, or rather the $\log \tan \frac{1}{2} E,$  that operation could be conveniently joined with the second, provided our table supplied directly the logarithm of the quantity

$$\frac{\tan \frac{1}{2} E}{\sqrt{ } A}$$

which differs from unity by a quantity of the second order. We have preferred, however, to arrange our table in a somewhat different manner, by which, notwithstanding the small extension, we have obtained a much more convenient interpolation. By writing, for the sake of brevity, $T$ instead of the $\tan ^{2} \frac{1}{2} E,$  the value of $A,$  given in article 37 ,

$$\frac{15(E-\sin E)}{9 E+\sin E}$$

is easily changed to