also, i'=l°46'58"-97 , n w n 



- Mr Sears C. Walker. 



vi THE THEORY OF THE LONG INEQUALITY 



The value of a de luced in the same way as a in the last Article, is 



a = 30-0363992, 



e = 0-00871946 J ' y 

 Solving the triangle PQR (fig. l), we shall find 



PQ = 56° 53' 54"- 15, 

 QR = 25 30 30-34, 

 PR = 82 23 48-14. 



Hence the quantity to be subtracted from e' and w' or PQ + QR - PR = 36"-35, and the 

 quantity to be added to SI ' in order to give IT, or PR - PQ, = 25° 29' 53"-99- 

 The values of the elements of Neptune, then, which will be employed, are, 

 Epoch Greenwich mean noon, January 1, 1847. 

 e' = 328° 32' 7"-85, 

 w' = 47 11 30-15, 

 e = 0-00871946, 

 n = 2l"-5549201 1, 

 a' = 30-0363992, 

 y' = 1° 30' 24"'64, 

 II' = 155°34'l4"-80, 



m = . 



18780 



7. Expanding the value of R given in (Art. 2), according to ascending powers of /, we 



have 



_ m'r ,„ „, m 



R = — cos (9 - 9') . 



r 2 yV + r'* - 2rr cos (0 - ff) 



m'r tn'rr'I 



+ ~7> {r 2 + r' 2 - 2rr' cos (0 - 0') J* ' &C " 



The developement of the first two terms was verified by comparison with the expansion 

 given by Pontecoulant (Theorie Analyt. du Syst. du Monde, Liv. VI). The developement of 

 the last two terms is, putting 



{a 2 + a' 2 -2aa'cos(0-0')}-i- \ = \B +... + B k cos k (0 - 0'), ' 



and £ = nt + e, £' = nt + e', 



/3 = nt + e — nr, /3' = n't + e — sr, 



R = -£ aa'S, shrl 7 'cos{&(£ - f) - (£ + f - 2D')} 



f 



til 



+ -aa^sin 2 l 7 'cos{(£ -!)(£-£')} 



