2] ON THE PERTURBATIONS OF URANUS. 15 



It will be seen that, in consequence of the arrangement which has been 

 given to the equations of condition, the equations thus formed naturally 

 separate themselves into two groups, one of which involves only Se, 8x l} 8x 2 , 

 with the quantities h and p, while the other involves Sn, Sy lt Sy, with the 

 quantities k and q. 



Also the coefficients in these equations are easily calculated by the 

 following formulae, putting t=10 in their right-hand members: 



. sin TO 



22 cos mt = 



sin m 



(t + 1) sin mt t sin m (t+l) 

 22 sin mt = - . .... 



. 

 2 sin 



1 (sin (m - n) (t + $). sin(m + n) (t + $j] 

 22 cos mt cos nt = - -.-- -, ,- \ ^ --- i / ^v 



2{ sin$(mn) sin^-(m + n) J 



, . 1 [sin (m -n)(t + i) sin(m + 



S2 sin mt sm nt = - . n/ - N --- ? r-r- r- 

 2 [ sin % (m n) sin -J (m + n) } 



1 lsinm(2<+l) 



22 COS 2 OT* = t + - + - A 



2 2 sm TO 



1 1 sin TO (2Z + 1 ) 

 S2sin s m =; + --- -^- ^. 



2 2 sin m 



15. By performing the calculations, the equations of the first group 

 are found to be the following : 



(e) 15l'-48= 21-0000 8e+ 6*06708^- 4'4358 &K, 

 + 13-6320 h,+ 0-4043 h. 2 - 4'5608 h.. 

 + 18-6046^+ 19-3384 p,+ 7'3721 _p 3 



(a;) 246-48= 6'06708e+ 8-28218^+ 4*1762 Ssc, 

 + 7-4041 A, + 8-2523 A 2 + 4'6963 A 3 

 + 6-5389^+ 6-3978 p,+ 8'1831 j> 3 



(A,) 209-74= 13-6320Se+ 7-40418^- 0'2337 Sx, 

 + 10-7022^+ 4-5356 ^ s - G'0018 h, 

 + 12-7013^ + 12-9883 p,+ 8'0038 ^ 8 



(h,) 242-68= 0-4043 8e+ 8'25238a^+ 7-5650Sa3 3 

 + 4-5356^ + 10-2960 h,+ 8'1944 h, 

 + 1-7866^+ 1-3667 p,+ 7'6671 p 3 



