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/41

 $$\begin{aligned} & e \sin \varepsilon=-29543^{\prime \prime} .18=-8^{\circ} 12^{\prime} 23^{\prime \prime} .18 \\ & M+e \sin \varepsilon \quad \text {. . . . } 3241631.59 \\ & \text { differing from } \varepsilon \text {. . . } 8.41 \text {. } \end{aligned}$$

This difference being multiplied by $\frac{\lambda}{\mu-\lambda}=\frac{29.25}{117.75}$ gives $2^{\prime \prime} .09,$  whence, finally, the corrected value of $E=324^{\circ} 16^{\prime} 31^{\prime \prime} .59-2^{\prime \prime} .09=324^{\circ} 16^{\prime} 29^{\prime \prime} .50,$  which is exact within $0^{\prime \prime} .01.$

14.
The equations of article 8 furnish several methods for deriving the true anomaly and the radius vector from the eccentric anomaly, the best of which we will explain.

I. By the common method $v$ is determined by equation VII., and afterwards $r$  by equation II.; the example of the preceding article treated in this way is as follows, retaining for $p$  the value given in article 10.

$$\begin{aligned} & \frac{1}{2} E=162^{\circ} 8^{\prime} 14^{\prime \prime} .75 \quad \log e \quad \text {. . . . } 9.3897262 \\ & \log \tan \frac{1}{2} E \text {. . . } 9.5082198 n \quad \log \cos v \text {. . . . } 9.8496597 \\ & \log \tan \left(45^{\circ}-\frac{1}{2} \varphi\right) \cdot 9.8912427 \quad 9.2393859 \\ & \log \tan \frac{1}{2} v. . . .9 .6169771 n \quad e \cos v \quad=0.1735345 \\ & \frac{1}{2} v=157^{\circ} 30^{\prime} 41^{\prime \prime} .50 \quad \log p \text {. . . . } 0.3954837 \\ & v=315 \quad 123.00 \quad \log (1+e \cos v) \cdot .0 .0694959 \\ & \log r \text {. . . . } 0.3259878 \text {. } \end{aligned}$$

II. The following method is shorter if several places are to be computed, for which the constant logarithms of the quantities $\sqrt{a(1+e)}, \sqrt{a(1-e)}$ should be computed once for all. By equations V. and VI. we have

$$\begin{aligned} & \sin \frac{1}{2} v \sqrt{ } r=\sin \frac{1}{2} E \sqrt{a(1+e)} \\ & \cos \frac{1}{2} v \sqrt{ } r=\cos \frac{1}{2} E \sqrt{a(1-e)} \end{aligned}$$

from which $\frac{1}{2} v$ and $\log \sqrt{ } r$  are easily determined. It is true in general that if we have $P \sin Q=A, P \cos Q=B, Q$ is obtained by means of the formula tan $Q=\frac{A}{B},$  and then $P$  by this, $P=\frac{A}{\sin Q},$  or by $P=\frac{B}{\cos Q}:$  it is preferable to use