32 



THE ORBIT OF NEPTUNE. 



- 20".536 sin (X X) 



0.007 sin (2 a/ 2 ;i o' + 

 + 0.530 sin ( a' + 2 a o) 



.059 sin (A, -co') 



.340 sin (2 X a o') 



+ .022 sin ( 51' + 3 a 2 o 

 . .007 sin (A/ + a 2 u) 



.002 sin (2 a, o o') 



ACTION OF SATURN. 



8 log r = 

 + 345 cos (a/ 



+ 10 cos ( 3,'+ 2 A w ) 



2 cos (* o') 



+ 3 cos (2 a' a, o') 



15. Perturbations of the elements. Collecting and adding up the coefficients 

 of all sines or cosines of the same angle in the perturbations, \ve find them as 

 below. For 2, and a, their values, I t and -n t, are substituted. We shall 

 first collect those terms which are developed as perturbations of the elements, 

 namely, the secular variations, and all terms in the action of Uranus in which 

 i' 2 i. We find them to be as follows : 



+ 125".67 sin (2 7' 7) 



0.42sin (27' I 2n) 

 -0.36 sin (2? l+n if) 



+ 0.14sin (2 1 I + Tt' TI) 



30".93 sin (4 Z' 2 7 TI) 

 + 8 .03 sin (4? 27 Tt 7 ) 



0.03sin(47' 27+7^ 27t) 



+ 2".62 sin (6 7' 3 7 2 7t) 



1. 37 sin (62' 37 rt n) 

 + 0.17sin(67' 37 2 71') 



+ 

 + 

 + 



2163".60sin(27' 7 TI) 

 14 1.69 sin (2 7' 7 TI') 



0.56 sin (27' 7+ TT' 2 ?t) 

 0.21 sin (27' 7+7t 2 7 t / ) 

 1 .08 sin (27' 7 +71 2 T) 

 O.OSsin (27' 7+Tt' 2r) 



= + 125".G7 cos (2 ?' 7) 



+ .42 cos (21' I 2 TT) 

 0.36 cos (27' I it + 71} 

 + 0.14cos (27' 7+ rf n) 



30".93 cos (4 7' 2 7 n) 

 + 8 .03 cos (4 7' 2 7 7t') 



.03 cos (4 7' 2 7+ it 2*) 



+ 2".62 cos (6 7' 3 7 2 n) 

 1.37 cos (6 7' 3 7 Tt' 7t) 

 + .17 cos (6 I 37 2 TI') 

 +0".003U 



a = 1232 cos (2 7' 7 n) 

 + 92 cos (2 7' 7 rf) 



+ 85 cos (4 I' 2 7 2 n) 

 44 cos (4 7' 27 T^ n) 

 + 6 cos (4 7' 2 7 27^) 



71".93 sin (4 7' 2 7 2 n) 



+ 38 .09 sin (4 7' 27 TI' 7t) 



4 .99 sin (4 7' 27 2 7t') 



+ 4".36sin (67' 37 3 TI) 

 -3.27 sin (6 7' 3 7 Tt 7 27t) 

 + .85 sin (G 3 7 2 rt n} 



0.08 sin (07' 37 3^) 



+ 2V.4425 i 



