334 



PROCEEDINGS OF THE AMERICAN ACADEMY 



" This table was computed from the following formulse for the 

 perturbations of the mean longitude and radius vector of Uranus, 

 which are arranged in a form similar to that proposed by Leverrier, 

 and adopted in his theory of Mercury. The mean longitude of each 

 planet is denoted by the appropriate symbol of the planet. The ele- 

 ments of Neptune which are adopted are those last given by Mr, 

 Walker, and the mass of Neptune which is introduced into the formulse 

 is 2-7j-^7j(j-th of the sun's mass, for which any other mass is readily sub- 

 stituted by simple multiplication. 



" The perturbation of the mean longitude z= dv = 



n. (9— ]^) — 0.02 cos. ( I? — f ) 



2 ( 51 — f ) — 0.99 cos. 2{w — '^) 



3 (11— f) — 0.01 cos. 3(¥ — f) 



4 (it— :^) 4-0.18 cos. 4(i[ — f) 



5 ( ^ _ :^) _ 0.01 cos. 5 ( IT — f ) 



6 (it— f) 4-0.20 sin. 7(11 — :^) 

 8 (n— f )-f0.04 sin. 9(^ — ]^) 



10 (ij_f)-j-0.01 sin. 11 (IT — f) 

 12(ii-f)-f^ 



+ 0'.00434 i sin. ( II — t^^j) — 0.03541 1 t cos. (^r — cj ) 



-j- k sin. ( II -h <5 — STjy) 

 -f- ki sin. (2 II -}- (9, — 2 f 

 + Z-'a sin. {W + d-z 

 in which 



¥■ 



cr 



¥ 



) 



J =2692.74 sin. (21^—11- 

 4- 106.80 sin. (4:^:— 2ii 



-2cr 



— 6.09 sin. 

 4- 0.48 sin. 



4-149'.'76 cos. 

 ^)— 43.08 cos. 



■2ii- 



'"m) 



¥ 



(2¥- 

 (4f_2ii-2^3^) 



(8f— 4ii — 4t^ )-f 0.47COS. (8 :^ — 4ii — 4 oTj^,) 



(6^—311 — 3^^)4- 4.20COS 



— 0.06sin.(10:^— 5ii — 5t^3^)4- 0.01 cos. (10 ]|: — oil — 5 stj^) 



^sin. ^=:— 2.58 sin. (:^ — :|:) — 0.44 cos. ( i^ — :^) 



— 11.35 sin. 2(11 — ]^)— 0.02 cos. 2(ii — ^i) 

 4- 18.27 sin. 3{-^—^)-}- 3.68 cos. 3 (11 — :^) 

 -1-66.38 sin. 4 ( 11 — :^) -f- 13.50 cos. 4(ii — f) 



— 2.74 sin. 5(11 — :^)— 0.56 cos. 5{w—'^) 

 ■— 0.78 sin. 6(11—]^)— 0.17 COS. 6(¥ — 1^) 



— 0.26 sin. 7(11 — ]|;)— 0.05 cos. 7(li— J^J) 



