Prof. P. E. Chase on Planetary Interaction. 509 



The difference between the actual value of log h f* vec. and 

 the value as found by the equation Y 1 x S 3 x^ 6 =h 10 ,is there- 

 fore only '000049, representing a numerical difference of only 

 ^Ly of 1 per cent. 



When the hypothetical nebular condensation had proceeded 

 so far as to show the controlling planetary influence of Jupi- 

 ter's mass, the mean perihelia of Saturn and Uranus were so 

 fixed as to establish the following relationships of harmonic 

 powers (mean perihelion, mean, and mean aphelion* being 

 designated by the subscript figures 1, 2, 3 respectively). 



Stockwell's logarithmic values are : — 



Neptune... *y 3 l-481951a ^ l-473327a' 



Uranus ... S x 1-262996/3 l 3 1-301989 & 



Saturn ... hi '957973 y ho '979496 7' 



Jupiter ... 2^3 -7345885 n 2 '716237 8' 



±(* -£)+i(/3'- r ) + (S -/) = -000085 = log 1-0002 



|( a / -/) + i(/3 / - r ) + (8 / - 7 ) = -000382=log 1-0009 



These results represent the following equations of distance : — 



©Mf-;)**©'-" (t)M?-;N!i)*-'- 



The theoretical differ from the actual values by less than -fa 

 of 1 per cent, in the first, and by less than ^ of 1 per cent, 

 in the second equation. The closeness of these agreements 

 may, perhaps, induce a glance at a few that are somewhat 

 more intricate. 



If we substitute for the theoretical primitive figurate expo- 

 nents (1, 3, 6, 10) the present actual vector radii (a=Y 2 ; 

 6=S 2 ; c=h 2 ; d=ii 2 )> we ma J form an equation for Saturn's 

 mean perihelion : — 



H^x$*- d x n 2 =h a ^ b (1) 



If a, b, d represent the mean aphelion vector radii, the equa- 

 tion represents Saturn's mean distance : — 



•yf xh b 3 ~ d X ^«=h? +6 (2) 



If we take powers of the masses instead of powers of the 

 vector radii, equation (2) yields two values for Saturn's mass, 

 according as we use Newcomb's greatest or least value of Nep- 

 tune's mass — 



the greatest (xgibo)? deduced from Y' s satellite, . . (3) 

 or the least (1797 od)> ?> •>•> perturbations of & . (4) 

 These four equations give approximations to precise accu- 

 * Smithsonian Contributions, 232, p. 88. 



