2] ON THE PERTURBATIONS OF URANUS. 27 



To these we may add the following, which are of two dimensions in 

 terms of the eccentricities : 



+ 6'-40 tri sin 3 {nt - n't + e - e'} 



- 074 m'e' sin {3 (nt - n't + e - e') - CT + or'}. 



34. Now, on our present assumption, 

 n=130'-6, w' = 448'-5, n-n' = 8 12''1, n-2n' = 3 23''6, 2n-3w'= 11 35'7. 



Hence the equations of condition given by the modern observations will 

 be of the form 



c= 



+ Sw + Sy 1 sin {13 0'5}t + 8y, sin {26 1-0} t 



+ h l coB{Bl2'l}i+ k, cos {16 24-2} t + h., cos {24 36 '3} t 

 + k 1 sm{ Sl2'l}t + k, sin {16 24'2} t + k 3 sin {24 36'3} t 

 + pi cos { 448'5}<+ _p., cos{ 3 23'6} t+p 3 cos{ll 357} t 

 + g,sin{ 448'5}+ g 2 sin{ 3 23-6}t + q 3 sin {11 357} . 



35. Treating these equations of condition in the same manner as before, 

 the equations in the first group, derived from them, are found to be the 

 following : 



(e) 151-48= 21-0000 Se + 6 '067080!,- 4'4358 &c, 



+ 13-9515 A,+ 0-9471 h,- 4'5965 h :t 



+ 18-3916^+19-6752 p a + 8'4184 p t 



(x) 246-48= 6'0670Se+ 8-28218^+ 4'1762 8x, 



+ 7-3540 /i,+ 8-3027 li.,+ 5'0961 h 3 



+ 6-5793^+ 6-3319 'p*+ 8'0850 p., 



(h,) 207-58= 13-9515 Se + 7-3540&C,- 0'4177 Sx, 



+ 10-9735 h,+ 4-6775 A 2 - O'OOOS /*, 



+ 12-8697^+13-4050 p.,+ 8"4781 > 



(A,) 245-17= 0-9471 8e+ 8 '30278^+ 7'2362 8a; 3 



+ 4-6775 7^+10-0259 7i 2 + 8'3220 A 3 



+ 2-3661^+ 1-6727 p 2 + 7'3073 p 3 

 103-48=- 4'5965Se+ 5'0961 8x, + 10'5558 8x 



- 0-0005 h,+ 8-3220 /,+ 10-9749 /t, 



- 2-8935 p,- 37316 jp 2 + 3'5852 p 3 . 



42 



