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/47

 But the Barkerian is by far the most convenient, and is also annexed to the admirable work of the celebrated OLBERS, (Abhandlung über die leichteste und bequemste Methode die Baln eines Cometen zu berechnen: Weimar, 1797.) It contains, under the title of the mean motion, the value of the expression $75 \tan \frac{1}{2} v+25$ $\tan ^{3} \frac{1}{3} v,$  for all true anomalies for every five minutes from 0 to $180^{\circ}.$  If therefore the time corresponding to the true anomaly $v$  is required, it will be necessary to divide the mean motion, taken from the table with the argument $v,$  by $\frac{150 k}{p^{\frac{3}{2}}},$  which quantity is called the mean daily motion; if on the contrary the true anomaly is to be computed from the time, the latter expressed in days will be multiplied by $\frac{150 k}{p^{\frac{3}{2}}},$  in order to get the mean motion, with which the corresponding anomaly may be taken from the table. It is further evident that the same mean motion and time taken negatively correspond to the negative value of the $v;$ the same table therefore answers equally for negative and positive anomalies. If in the place of $p,$ we prefer to use the perihelion distance $\frac{1}{2} p=q,$  the mean daily motion is expressed by $\frac{k \sqrt{ } 2812.5}{q^{\frac{3}{2}}},$  in which the constant factor $k \sqrt{ } 2812.5=$  0.912279061, and its logarithm is 9.9601277069. The anomaly $v$ being found, the radius vector will be determined by means of the formula already given,

$$r=\frac{q}{\cos ^{2} \frac{1}{2} v}$$

20.
By the differentiation of the equation

$$\tan \frac{1}{2} v+\frac{1}{3} \tan ^{8} \frac{1}{2} v=2 t k p^{-\frac{8}{2}},$$

if all the quantities $v, t, p,$ are regarded as variable, we have

$$ \frac { \mathrm{d} v}{2 \cos ^{4} \frac{1}{2} v}=2 k p^{-\frac{8}{2}} \mathrm{~d} t-3 t k p^{-\frac{8}{2}} \mathrm{~d} p $$

$$ \mathrm{~d} v=\frac{k \sqrt{ } p}{r r} \mathrm{~d} t-\frac{3 t k}{2 r r \sqrt{ } p} \mathrm{~d} p$$