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/37

 Given $v=310^{\circ} 55^{\prime} 29^{\prime \prime} .64$, $\varphi=14^{\circ} 12^{\prime} 1^{\prime \prime} .87$ , $\log r=0.3307640$ ; $p,$ $a,$  $E,$  $M,$  are required. [The letter $n$ affixed to a logarithm signifies that the number corresponding to it is negative.]

hence $\tfrac{1}{2}(v-E) = -4^{\circ}58^{\prime}22^{\prime\prime}.94;$ $v-E=-9^{\circ}56^{\prime}45^{\prime\prime};$  $E=320^{\circ}52^{\prime}15^{\prime\prime}.52.$

Further, we have

hence $e\sin E$ in seconds $=31932^{\prime\prime}.14=8^{\circ}52^{\prime}12^{\prime \prime} .14;$  and $M=329^{\circ} 44^{\prime} 27^{\prime \prime} .66.$

The computation of $E$ by formula VII. would be as follows:

$$ \begin{array}{l} \frac{1}{2} v=155^{\circ} 27^{\prime} 44^{\prime \prime} .82 \\ 45^{\circ}-\tfrac{1}{2} \varphi=37^{\circ} 53^{\prime} 59^{\prime \prime} .065 \\ \vphantom{\frac{1}{2}} \end{array} \qquad \begin{array}{ll} \log \tan \frac{1}{2} v \quad.\quad.\quad. & 9.6594579 n \\ \log \tan (45^{\circ}-\tfrac{1}{2} \varphi) \;\;. & 9.8912427 \\ \hline \log \tan \frac{1}{2} E \;\;\;.\quad.\quad. & 9.5507006 n \end{array}$$

whence $\frac{1}{2} E=160^{\circ} 26^{\prime} 7^{\prime \prime} .76,$ and $E=320^{\circ} 52^{\prime} 15^{\prime \prime} .52,$  as above.