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/65

 $$\int \frac{p p \mathrm{~d} v}{(1+e \cos v)^{2}}: \int \frac{4 q q \mathrm{~d} w}{(1+\cos v)^{2}}=\sqrt{ } p: \sqrt{ } 2 q$$

the integrals commencing from $v=0$ and $v=0,$  or

$$\int \frac{(1+e)^{\frac{3}{2}} d v}{(1+e \cos v)^{2} \sqrt{2}}=\int \frac{2 d w}{(1+\cos w)^{2}}$$

Denoting $\frac{i-e}{1+e}$ by $\alpha, \tan \frac{1}{2} v$  by $\theta,$  the former integral is found to be

$$\sqrt{ }(1+\alpha) \cdot\left(\theta+\frac{1}{3} \theta^{3}(1-2 \alpha)-\frac{1}{5} \theta^{5}(2 \alpha-3 \alpha \alpha)+\frac{1}{7} \theta^{7}\left(3 \alpha \alpha-4 \alpha^{3}\right)-\text { etc. }\right)$$

the latter, $\tan \frac{1}{2} w+\frac{1}{3} \tan ^{3} \frac{1}{2} v.$ From this equation it is easy to determine $v$  by $\alpha$  and $v,$  and also $v$  by $\alpha$  and $w$  by means of infinite series: instead of $\alpha$  may be introduced, if preferred,

$$1-e=\frac{2 \alpha}{1+\alpha}=\delta$$

Since evidently for $\alpha=0,$ or $\delta=0,$  we have $v=w,$  these series will have the following form :-

$$\begin{aligned} & v=v+\delta v^{\prime}+\delta \delta v^{\prime \prime}+\delta^{3} v^{\prime \prime \prime}+\text { etc. } \\ & v=v+\delta w^{\prime}+\delta \delta v^{\prime \prime}+\delta^{3} v^{\prime \prime \prime}+\text { etc. } \end{aligned}$$

where $v^{\prime}, v^{\prime \prime}, v^{\prime \prime},$ etc. will be functions of $v,$  and $v^{\prime}, v^{\prime \prime}, w^{\prime \prime \prime},$  functions of $w.$  When $\delta$  is a very small quantity, these series converge rapidly, and few terms suffice for the determination of $v$  from $v,$  or of $v$  from $w. \quad t$ is derived from $v,$  or $w$  from $t,$  by the method we have explained above for the parabolic motion.

35.
Our Bessel has developed the analytical expressions of the three first coeffcients of the second series $w^{\prime}, w^{\prime \prime}, w^{\prime \prime \prime},$ and at the same time has added a table constructed with a single argument $w$  for the numerical values of the two first $v^{\prime}$  and $u^{\prime \prime},$  (Von Zuch Monatliche Correspondena, vol. XII., p. 197). A table for the first coefficient $w^{\prime},$ computed by Simpson, was already in existence, and was annexed to the work of the illustrious OLBERs above commended. By the use of this method, with the help of BeSSEL’s table, it is possible in most cases to determine the true anomaly from the time with sufficient precision; what remains to be desired is reduced to nearly the following particulars: