MEAN DISTANCES OF THE PLANETS. 239 



that such secondary causes are at work. But at present the 

 astronomer is fully justified in asserting that the mean distances 

 of the planets (semi-axis-major of the elliptical orhit) is in- 

 variable. In a paper published in this journal in 191 1. January, 

 the writer gave a table of the mean distances of the eight major 

 planets, but he remarked that the values given for the planets 

 Uranus and Neptune were not the true mean distances as they 

 had been derived from mean motions which included effects due 

 to Newcomb's method of dealing with the great inequalitv in 

 the motions of those two planets. This paper will give revised 

 figures and will include the mean distances of the two small 

 planets Eros and Ceres. 



The writer is of the opinion of the American astronomers, 

 Hill and Newcomb, and of the French astronomer Tisserand 

 (who adopts Laplace's procedure) that the mean distance of 

 a planet should be so chosen that it is only subject to periodical 

 variations. Unfortunately it is very difficult to measure the 

 distance of a planet from the Sun directly, and astronomers 

 always use an indirect method founded upon Kepler's law con- 

 necting the distances and periods. If there were but one planet 

 and the Sun. the equation connecting the two quantities is 



a3n- = k-{i -\- 111) 

 in which 



a = semi-axis major (mean distance). 



n = mean motion in a fixed unit of time. 



k = attractive force of the Sun in some convenient unit. 



ui = mass of the planet in terms of the mass of the Sun. 

 If another planet is considered ( let us suppose it to be the 

 Earth), and the mutual attraction of the two planets is left on 

 one side for the moment, we should have the similar equation 

 rt, 3 fi^2 = ^.2 ^j _|_ j,i y 



If we now choose that the mean distance of the Earth be the unit 

 ol distance, the distances of other planets can be determined 

 wdien their mean motions and masses are known. But the 

 above simple equations do not hold when the mutual attractions 

 of the planets are considered. The observed mean motion of 

 a planet is made up of one part (by far the greater) due to 

 the action of the Sun and a smaller" part due to the action of the 

 other planets of the system. The same remarks apply to a 

 planet's mean distance. The full elucidation of this problem 

 depends on the disposal of the constants introduced by the 

 integration of the differential equations of motion. The net 

 result is that if the action of the planets increases the motion 

 of any planet by an {cr being a very small factor in the Solar 

 System) the effect on the mean distance computed by means of 

 the above equations is lo-. If the constants added to the mean 

 distance itself are neglected, this correction would be fo-, which 

 is four times too great. Reference to Tisserand's Mecanique 

 Celeste, t. iv, p. 18, should be made by those wishing to find 

 proofs of these statements. The formulge which are applicable 



