32 THE ORBIT OF NEPTUNE. 



ACTION OP SATURN. 



Sv=: S log r = 



— 20".536 sin {?,' — X) + 345 cos {?,' — X) 



— O.OOTsin (2;i' — 2;i — w' + o) 



+ .530 sin (— :i' + 2 A — cj) + 10 cos (— ;i' + 2 ;i — w) 



— .059 sin (;i — o') — 2 cos (X, — u') 



— .340sin (2;i' — ;i — o') + 3 cos (2 ;i' — ;i — w') 

 + .022 sm{—X-\-S^ — 2u) 



— 0.007sin(;i' + ;i — 2 a)) 



— 0.002 sin {2^ — ci — u') 



§ 15. Perturbations of the elements. — Collecting and adding np the coefficients 

 of all sines or cosines of the same angle in the j^erturbations, we find them as 

 below. For "k and o, 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 

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



hh = + 125".67 sin (2 1' — I) hk - + 125".67 cos (2 1' — Z) 



— .42 sin [21' — 1 — 2 n) + .42 cos {21' — 1 — 2 n) 



— 0.36 sin (2Z' — Z+7t — 7t') _0 .36 cos (2 Z' — Z — 7t'+ n) 

 + .14 sin (2 l' — l+ n'— n) +0 .14 cos (2 Z' — Z + «' — 7t) 



— 30".93 sin (4 Z' — 2 Z — 7t) — 30".93 cos (4 Z' — 2 Z — 7t) 

 + 8.03sin (4Z' — 2Z — Tt-) + 8 .03 cos (4Z' — 2 Z — 7t') 



— 0.03sin(4Z' — 2Z+7t' — 2 7t) — .03 cos (4 Z'— 2 Z+ 7t'— 2 7r) 



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



— 1 .37 sin (6 Z' — 3 Z — ti' — n) — 1 .37 cos (6 Z' — 3 Z — n' — n) 

 + .17 sin (6 Z'— 3 Z — 2 7t') +0 .17 cos (6 Z' — 3 Z — 2 n') 



+ 0".0132 t + 0".0031 1 



U = + 2163".60sin (2Z' — Z — 7t) Moga = — 1232cos (2 Z' — Z — jt) 



— 141 .69 sin (2 Z' — Z — 7t') + 92 cos (2 Z' — Z — ti') 

 + 0.56sin(2Z' — Z+7t' — 2 7t) 



+ 0.21sin(2Z' — Z+7t— 2 7t') + 85cos (4 Z' — 2 Z — 2 7t) 



+ 1 .08sin (2Z' — Z+ 7t — 2t) — 44 cos (4 Z'— 2 Z — ti' — 7t) 



— O.OSsin (2 Z' — Z +7t' — 2 t) + 6 cos (4 Z' — 2 Z— 2 7t') 



— 71".9 3 sin (4 Z' — 2 Z — 2 n) 



+ 38 .09 sin (4 Z' — 2 Z — 7t' — 7t) 



— 4. 99 sin (4Z' — 2Z — 27t') 



+ 4".36sin (6Z' — 3Z— 3 7t) 



— 3 .27 sin (6 Z' — 3 Z — 7t' — 2 7t) 

 + .85 sin (6 Z' — 3 Z — 2 7t' — 7t) 



— O.OBsin (6?— 3Z— 3 7t') 



+ 21".4425 t 



