THE ORBIT OF NEPTUNE. I7 



B''^ = u ( — j^i — vii) 



BP'^^udi— ^i_|r,+ Vi5 — i-i'2i) (19) 



£W = «e( 21^3 — 2x'22) 



The values of iV'"' are as follows : 



Nm - 



1' — % 



i\r(3) - 



— ;i' + 2 ;i — G) 



Ni^) — 



2W — 7. —6)' 



NO) - 



;i + cj' — 2 



i\^(«) = _ X' + 3 a — 2 CO 



iV^Wzr 2 7,— a —d 



N<^^^= 3 7J — 7, —2(0' 



iVrm- 2:i' + ^ _o' — 2co 



■i\^(i^)=: 2;i' + ;i —3 a)' 



jVr(2i)- 2;i' + X — w' 



iv^('' = ;i' + A —6)' 



jVr(2) _ 



2;i' — 2;i — G)' + u 



i\r(4) — 



/I — w' 



i\r(6) -_ 



-2;i'+3;i + w' — 26) 



i\r(8) - 



3;i'_2X — 2a)' + G) 



j\r(io) i:^ 



;i' + ;i — 2 



j\r(i2) - 



;i' + ;i — 2 0' 



^(11) _ 



;i + 0' (20) 



jy(i6) _ 



3 ^ — 0)' — 2 



_/V(18) — . 



/I' + 2 /I — 2 a)' — a) 



_^(20) _ 



4;i' — ^ —3 a)' 



N(-) - 



^ 



N^c) _ 



X' /I 6)' 



i\^<'" zz 2;i— 0) 



From these values of N the corresponding values of v are derived, remembering 



that 



n' 

 i'n'-{-in 



i' and i being the coefficients of TJ and X respectively in the value of N. 



The check on the correctness of the preceding values of V, R, arid B may now 



• ??/ 



be applied by developing v in powers of — , and retaining only the first term ; 



that is, by putting i; = -^, v^ = 0. Making these substitutions, all the values of 



V, R, and B will be found to vanish. In other words, |U^ will be the lowest power 

 of (U which will enter into the values of V, R, or B, as we have already shown from 

 a priori considerations. 



For convenience, we shall give the values of V, R, and B developed according to 

 the powers of n, the ratio of the mean motions, a form similar to that in Avhich 

 the lunar inequalities are developed in the theory of the moon. Putting 



we have 



V -^{1— s^+ sY— sy + etc.} 

 v-' = ^{l — 2s^ + ?, sy — 4 sY + etc.} 



May, 1865. 



