JUPITER AND SATURN: IXj. 



L.V.'s &i^"'= 0-4407620+ li -29503 X 0-0004301 + ( 4-96 j 

 + _ 5575 

 0-4413195 in perfect agreement. 

 L.V.'sV"" = 0-0100473 + 0-1055 X etc. 

 454 



0*010093 where Le Verrier has 0-010083 ! i^'' this, 

 case Le Verrier is undoubtedly in error, because we iind that 

 Runkle's tables for log 0=9-7367410 give 0-010096.* 



It is not without interest to compare the secular part of the 

 elliptic disturbing function and this is done in the annexed table. f 

 The first, third and fifth figure-columns give the values of the 

 coefficients for log a = 9-7367408, the second, fourth and sixth 

 for log a = 97365540. Logarithms of the numbers are given. 



■The columns headed a' Rj, a' a -^—^ etc., show Le Verrier's figures ; 



a a 



those headed F, DF, etc., the new figures. As will be gathered 

 a'Ri = F, the latter being Newcomb's symbol. To get a' a — — ^ 



etc., Le Verrier has had a long and arduous work, which is evaded 

 by using the " D " operator. Many of the coefficients are closely 

 related and afford immediate checks, which however were only 

 very incompletely known to Le Verrier. Thus {b) — 4 (a), a. 

 condition only satisfied by one pair of Le Verrier's figures ; (g) = 

 4 (e), (j) = 8 (/), and (/) = 8 (d), conditions which are also not 

 rigorously fulfilled. Then the calculation of many of the co- 

 efficients can be greatly abbreviated on the forms of both Newcomb 

 and Le Verrier, thus (a) = ^ &i 3 ; (c) = ^ b-i 3 ; (e) = — ^2 W » '■> 

 [i) = — ^:^ ^2. 5 ; (^) = f h. 5 ; ("^) = — IM ^0, 7 ; to quote only a, 

 few. The same simplifications hold with D and (i 4- D) which is 

 indeed obvious enough in Newcomb's notation, but it is hidden 

 in Le Verrier's . Le Verrier's second and third columns for the- 

 coefficients (h) are entirely erroneous. 



Secular Inequalities. 



The effect of the second and higher powers of the masses so- 

 modifies all but the lowest terms of the secular variations that sl 

 comparison of the lowest terms will suffice. The terms added by 

 the higher powers of the excentricities are in several cases increased 

 100-fold by including the second powers of the masses in the mutual 

 perturbations of Jupiter and Saturn. 



Secular variation of the epoch : — this is not of much importance- 



in the present theory ; we have 



de 

 dt 



Le Verrier Term in 



— 7-46858 — 7-48126 



— 0-14440 — 0-14476 e-2+e,2 



-t- 0-01270 -I- 0-01273 no 



+ 0-02399 +0-02406 ee, cos (w' — w). 



* Runkle, New Tables, p. 54. 

 -f See next page. 



