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/79

 $$\begin{aligned} & \log \alpha=9.9758345 \\ & \log \beta=9.0251649 \\ & \log \gamma=9.9807646 \end{aligned}$$

Next we have $\log \alpha t=1.7914943,$ whence by Barker’s table the approximate value of $w=70^{\circ} 31^{\prime} 44^{\prime \prime},$  and hence $A=0.052983.$  To this $A$  in our table answers $\log B=0.0000207;$  from which, $\log \frac{\alpha t}{B}=1.7914736,$  and the corrected value of $w=70^{\circ} 31^{\prime} 36^{\prime \prime} .86.$  The remaining operations of the calculation are as follows:-

$$2 \log \tan \frac{1}{2} w \quad \text {. . } 9.6989398 \quad \log \tan \frac{1}{3} w \quad \text {. . . . } 9.8494699$$

$\log \beta.$. . . . $9.0251649 \log \gamma.$. . . . . 9.9807646

$\log A.$. . 8.7241047 C $\log \left(1+\frac{4}{5} A+C\right) \cdot 9.9909602$

$A=.$. . . . $0.05297911 \log \tan \frac{1}{2} v.$. . . 9.8211947

$\log B$ as before, $\quad \frac{1}{3} v=.$. $33^{\circ} 31^{\prime} 30^{\prime \prime} .02$

$$\begin{aligned} & C=. .0 .0001252 \quad v=\quad. \quad .6730 .04 \\ & 1+\frac{4}{5} A+C=\text {. } 1.0425085 \quad \log q \text {. . . . . . } 0.0201657 \\ & 1-\frac{1}{5} A+C=. .0 .9895294 \quad 2 \text { C. } \log \cos \frac{1}{2} v \quad. . .0 .1580378 \\ & \log \left(1+\frac{4}{5} A+C\right) \quad. \quad .0 .0180796 \\ & \text { C. } \log \left(1-\frac{1}{5} A+C\right) \cdot \cdot 0.0045713 \\ & \log r \text {. . . . . . . } 0.2008544 \end{aligned}$$

Those which we found above (article 26), $v=67^{\circ} 2^{\prime} 59^{\prime \prime} .78, \log r=0.2008541,$ are less exact; and $v$  should properly have resulted $=67^{\circ} 3^{\prime} 0^{\prime \prime} .00,$  with which assumed value, the value of $t$  had been computed by means of the larger tables.