29 



No. 38. The five normal places selected as the basis of the first approxima- 

 tion to the elliptic elements of Neptune were reduced to geocentric latitudes 

 and longitudes, then to heliocentric longitudes and latitudes, by assumed values 

 of r and of sec. (f3 = the hel. lat.,) with coefficients for d r. That of sec. p is 

 well enough known. The values of 6 1) and s »•, in Prof Peirce's 1st *Ephemeris, 

 were applied inversely to obtain the elliptic heliocentric longitudes and latitudes. 

 The new values of the ^ and i were derived from the two extreme normal 

 places ; hence were obtained the following data for «', or the true longitude on 

 the orbit : 



FIRST HELIOCENTRIC NORMAL PLACES. 



I ; ii = Jan. 1, 1847 — 18864'' = May 9, 1795 

 II ; 4 = — 134 = Aug. 30, 1846 



III ; /, = — 55 = Nov. 7, 1846 



IV ; t,= + 95 = April 6, 1847 

 V ; /,, = + 333 = Aug. 33, 1847 



= 315 48 7.68 — 44.1 (n — 30.38778) 

 = 326 45 30.83 — 1.2 (r„ — 39.99356) 

 = 327 13 58.57 — 227.6 (r, — 39.99356) 

 = 32S 8 0.67 + 163.8 (r, — 39.99256) 

 = 338 57 44.39 -|- 1.0 \r, — 29.99356) 



The value 29.99256, for the last four dates, was that value which for d r = o 

 would give ''-j^" = V = 21".61735. The formula -^ = (7 -^) ' gives for 

 these values of r and v, the results ^, = 21''. 55448, a = 30.03634, t = 164 ,^^^ = 

 period in tropical years. 



No. 39. This value of ^ is a deduction from direct observation, and has in it 

 no other source of theoretical error than that which arises from the change of 

 the radius vector in 150 days, and from the change of ,, in the interval of 88 

 days from April 30, 1847, backwards, to Jan. 20. In order to estimate this 

 source of error, I shall here anticipate results not known in this stage of the 

 inquiry. We have, by Elliptic Elements I, the values 



Atv= —0". 0000271 

 At ^ = + 0". 0000335 



The small modification of ^ derived from the use of these small corrections, 

 was not found, on subsequent trial, to give better elements than the above value 

 of ^ = 21". 55448. After making several approximations towards the Elhptic 

 Elements, the radius vector was found to diminish from 29.99256, January 20, 

 1847, to 29.98676, August 22, of that year. Hence, the fundamental quantities 

 for making the approximations towards the true value of the elements, are, 



o I II II II 



May 9, 1795, n + v, = «, = 215 48 7.68 — 44. 1 (r, — 30.38778) 

 August 32, 1847, Tt + v.^ u>,= 338 57 44.39 + 1.0 (r, — 29.98676) 

 ft = 21.55448 



- See Proceedings R. A. S., Vol. VIII, p. 40. 



