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/70

 $17^{\circ} 22^{\prime} 38^{\prime \prime} .64.$ To this value of $E$  corresponds $\log B=0.0000040;$  next is found in parts of the radius, $E=0.3032928, \sin E=0.2986643,$  whence $\frac{9}{20} E+\frac{1}{20} \sin E$  $=0.1514150,$  the logarithm of which $=9.1801689,$  and so $\log A^{\frac{1}{2}}=9.1801649.$  Thence is derived, by means of formula [1] of the preceding article,

$$\begin{aligned} & \log \frac{2 B q^{\frac{3}{2}}}{k \sqrt{ }(1-e)} \cdot .2 .4589614 \log \frac{2 B(1+9 e)}{15 k}\left(\frac{q}{1-e}\right)^{\frac{3}{2}} \cdot 2.7601038 \\ & \log A^{\frac{1}{2}} \text {. . . . } 9.1801649 \log A^{\frac{3}{2}} \text {. . . . . . . . } 7.5404947 \\ & \log 43.56386=. \quad 1.6391263 \log 19.98014=. . . . .1 .3005985 \\ & 19.98014 \\ & \overline{63.54400}=t \text {. } \end{aligned}$$

If the same example is treated according to the common method, $e \sin E$ in seconds is found $=59610^{\prime \prime} .79=16^{\circ} 33^{\prime} 30^{\prime \prime} .79,$  whence the mean anomaly $=$  $49^{\prime} 7^{\prime \prime} .85=2947^{\prime \prime} .85.$  And hence from

$$\log k\left(\frac{1-e}{q}\right)^{\frac{3}{2}}=1.6664302$$

is derived $t=63.54410.$ The difference, which is here only $\frac{1}{0} \frac{1}{00}$  part of a day, might, by the errors concurring, easily come out three or four times greater. It is further evident, that with the help of such a table for $\log B$ even the inverse problem can be solved with all accuracy, $E$  being determined by repeated trials, so that the value of $t$  calculated from it may agree with the proposed value. But this operation would be very troublesome: on account of which, we will now show how an auxiliary table may be much more conveniently arranged, indefinite trials be altogether avoided, and the whole calculation reduced to a numerical operation in the highest degree neat and expeditious, which seems to leave nothing to be desired.

39.

It is obvious that almost one half the labor which those trials would require, could be saved, if there were a table so arranged that $\log B$ could be immediately taken out with the argument $A.$  Three operations would then remain; the first indirect, namely, the determination of $A$  so as to satisfy the equation