16 



THE ORBIT OF NEPTUNE. 



§ 10. That portion of the perturbative function which is due to the action of 

 the inner planet on the sun is 



^cos V 



r- 



F being the angular distance between the planets. Developing it in a series of 

 terms of the form 



— ;- cos (iT + i'^ +/"' -^3^) 



h will be of the form -^, c being a numerical coefficient, multiplied by powers of 



the eccentricities and mutual inclinations. 



From this development, and the equations (3), (4), (5), (7), and (8), I have 

 computed the following analytical values of the coefficients for the perturbations 

 of the longitude, latitude, and logarithm of radius vector. 



h 



--.1. F«siniV« 

 a} 



h log ; 



a: 



h^ 



= -* 2 B^ sin iV(«> 



(16) 



FW =r (1 — iC' — \ r) ijW + 2j^i + 3i., + V,) 



+ 6'^ (— fj^r — fj^i + Sva-— 2r,— 1^12+ Vna) 

 F(-) —ed{ 6 vi + I x'2 + 6 vi— 3 rg + 3 i^s + V' ^n) 

 F'> -e (— 6^3^ + Iv; — 2v.,— Gi'n) 



FW =d (— Zv,'— 1^1 — f T',— ^v, + \v,,) 



V('> =(^ ( 3v;^+ ^v, + 3i;/+ ^v,+\^v,+ |ri3) (17) 



|r(9) _ g2 ( 2„7 ^^ _ y ^^2 _|_ 2^7 ^^ _ 8j ^^^) 



F(12) =^^—l^ V,' — iv, — V ^4 — V '^5 + f ^r/ + i ^^12 + i ^lo) 



y(13) _ ^ ( 1^5 ^^2 ^ 1^5 ,,^ + 3_0 ^^ + 3 ^^2 + I ^^ + 2^7 ^^^2 _|_ I ^^3 + 2 7.,o) 



^(15) _ ,,2 ( 3 ^^^,_^ 2 V^, — 3 T'l, + Z^,i) 



E(^) zz (1 — w^ — 1 e") (_ 2 1^1 — I r^ + i rs) ■ • 



JBW zze' 



jjao) _ e2 

 i2f'2> — e'2 



Vi 3^3"+ I 1^3 + 



^2 + 4 ^3 — 3 Vn) 



1 1^1^+ i^'i + f ^4 + 



I ri' — 1 1^1 — I x'i — 



2 7,, _L 2 7 ,,9 S 1 i> ^ 



16 ^6 -T 4 ^ 16 ^isj 



A ^7 ~ i ^^10 + iV J'n) 



|a,^ + 3^3^— I 





i^l2) 



(18) 



^ 1^3 + I r„ + 1 T'ls) 

 + ll ^4 + 11 ^5 — I J^12 + \ Vw) 

 ^1 —U'^i — I ^s" — tV ^6 — V ^'13 + ^20) 

 ^'15 + i 1^2l) 



