8 RELATIONS PERTAINING SIMPLY [BOOK I. 



number 206264.81. We can dispense with the multiplication by the last quan 

 tity, if we employ directly the quantity k expressed in seconds, and thus put, 

 instead of the value before given, k = 3548&quot;.18761, of which the logarithm = 



3.5500065746. The quantity - a expressed in this manner is called the 



a? 

 mean anomaly, which therefore increases in the ratio of the time, and indeed every 



day by the increment 7~ , called the mean daily motion. We shall denote 



a* 



the mean anomaly by M. 



7. 



Thus, then, at the perihelion, the true anomaly, the eccentric anomaly, and the 

 mean anomaly are = ; 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; but from this point to 

 the perihelion, the eccentric is alwa} r s greater than the true, and the mean 

 greater than the eccentric, until in the perihelion all three become = 360, 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 --E and the mean 360 M correspond to the true 

 360 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 to 360 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 by the mean daily motion -^ p^, from which it is apparent, that for dif- 



a 



ferent bodies revolving about the sun, the squares of the periodic times are pro 

 portional 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. 



