SUPPLEMENT. 



It now remains to take from these Tables the especial values of b^ and its deriva- 

 tives, which are needed in the perturbative theories of the planetary motions. The 

 present Supplement will only include those coefficients which pertain to the prin- 

 cipal Planets ; those for the Asteroids being reserved for a future occasion. 



In determining the values of a, the following values of the masses and mean 

 motions, which are those adopted in the American Ephemeris and Nautical Alma- 

 nac, will be used. 



Mercury, 



Venus, 



The Earth, 



Mars, 



Jupiter, 



Saturn, 



Uranus, 



Neptune, 



I 



390000 



1 

 354936 



1 



2680637 



1 



3501.6 



1 

 24905 



= 5381016.218 

 = 2106641.438 

 = 1295977.440 

 = 689051.030 

 = 109256.719 

 = 43996.127 

 = 15424.5094 

 = 7872.77382 



We have, then, for the value of a, the ratio of the mean distances of any two 

 planets, as Mercury and Venus, the formula 





which takes into account the correction, necessary in the perturbative theory, due to 

 the masses. 



But if each planet is considered with reference to the Earth, the values of the 

 a , &c, may first be found, since a m = 1, and then the 



mean distances a 1 , a) 



values of a. 



We find for 



Mercury, 



Venus, 



The Earth, 



Mars, 



Jupiter, 



Saturn, 



Uranus, 



Neptune, 



0.38709870 



log 



a 1 



= 



9.58782172 



0.72333227 



log 



a" 



= 



9.85933784 



1.00000000 



log 



a m 



= 



0.00000000 



1.52369140 



log 



a lv 



= 



0.18289702 



5.20280136 



log 



a v 



= 



0.71623725 



9.53885533 



log 



a vi 



= 



0.97949626 



19.18357126 



log 



a v " 



= 



1.28292946 



30.03680569 



log 



a v.n 



= 



1.47765375 



