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/34

 number $206264.81$. We can dispense with the multiplication by the last quantity, if we employ directly the quantity $k$ expressed in seconds, and thus put, instead of the value before given, $k=3548^{\prime \prime}{.}18761,$  of which the logarithm $=3.5500065746$. The quantity $\frac{k t \sqrt{1+\mu}}{a^{\frac{3}{2}}}$ expressed in this manner is called the mean anomaly, which therefore increases in the ratio of the time, and indeed every day by the increment $\frac{k \sqrt{1+\mu}}{a^{\frac{3}{2}}},$  called the mean daily motion. We shall denote the mean anomaly by $M.$

7.

Thus, then, at the perihelion, the true anomaly, the eccentric anomaly, and the mean anomaly are $=0;$ after that, the true anomaly increasing, the eccentric and mean are augmented also, but in such a way that the eccentric continues to be less than the true, and the mean less than the eccentric up to the aphelion, where all three become at the same time $=180^{\circ};$  but from this point to the perihelion, the eccentric is always greater than the true, and the mean greater than the eccentric, until in the perihelion all three become $=360^{\circ},$  or, which amounts to the same thing, all are again $=0.$  And, in general, it is evident that if the eccentric $E$  and the mean $M$  answer to the true anomaly $v,$  then the eccentric $360^{\circ}-E$  and the mean $360^{\circ}-M$  correspond to the true $360^{\circ}-v.$  The difference between the true and mean anomalies, $v-M,$  is called the equation of the centre, which, consequently, is positive from the perihelion to the aphelion, is negative from the aphelion to the perihelion, and at the perihelion and aphelion vanishes. Since, therefore, $v$ and $M$  run through an entire circle from $0$  to $360^{\circ}$  in the same time, the time of a single revolution, which is also called the periodic time, is obtained, expressed in days, by dividing $360^{\circ}$  by the mean daily motion $\frac{k \sqrt{1+\mu}}{a^{\frac{3}{2}}},$  from which it is apparent, that for different bodies revolving about the sun, the squares of the periodic times are proportional to the cubes of the mean distances, so far as the masses of the bodies, or rather the inequality of their masses, can be neglected.