OP URANUS AND NEPTUNE. li 



Ae' = + 89"-999 sin X' - 83"-276 cos X' 

 + 24"'981 sin 2X' - 23''-490 cos 2X' 

 + 2"-451 sin 3X' - 2" -406 cos 3X' 

 = + 102"-292. 



e'Aw = + 8S"-024 sin X' + 90"'762 cos X' 



+ 23" -501 sin 2X' + 24"'981 cos 2X' 



+ 2" -406 sin 3X' + 2" -451 cos 3X' 



= + 82"-322. 



To the above value of Ae may here be added the variation of e due to secular perturbation 

 during the 46 years between 1801 and 1847, which was not taken account of in Art. (5), and 

 which = — 2""51. Again the value of the annual increase of sr including the secular perturba- 

 tion produced by Neptune, and employing Bessel's value of precession, = 53". 115, which differs 

 from the value used in Art. (5) by + 0"-6l5 ; and this difference amounts in 46 years to 

 + 28"'29. The values, therefore, of the corrections will be, finally, 



Ae = - 103"-48, A-ar = + 2° 26' 4"-8, 



Ae'= + l02"-29, Aw' = + 2° 37' 2l"-0. 



Hence for the corrected values of the elements, we have 



log e = 2-6644323, log e'= 3-9645133, 



log sin or = T-2118325, log cos -er = T'994l623 -, 



log sin -sr^ T-8830684, log cos -za-' = 1-8097403. 



Upon recalculating those terms of S (nt + e) and I (n't + e), which depend upon the 

 first powers of the eccentricities, with these corrected values of the elements, we shall find that 

 the following terms must be added to those given in Art. (38.) 



# (nt + e) m + 7"'159 sin X - 13l"'76l cosX, 



# (n't + e) = - 5"-133 sin X + 94" -505 cos X, 



and the corresponding corrections of $e, eSw, Se, e'^Tir', are 



%e = - 17"*45 sin X + s"-26 cos X, 



e%Tsr = - 7"74 sin X - 16"-71 cos X, 



5V= - 3"-70 sin X - 4"-18 cosX, 



e'3V= + 8" -41 sinX+ l"-31 cosX. 



Again, An = - o"-00214598 at 1801, 



A»'= +0"-002268l6 at 1847. 



Hence w + Aw = - 0"-8834086. 



3m' (« + Aw) 2 



Therefore upon recalculating the principal terms, using the multiplier . 8 . -„ 



1 find that the following terms must be a( 

 y (nt + e) - + 4S"'74 sin X - 4"-53 cosX, 



3m' w 2 



instead of — — — Tl we shall find that the following terms must be added, 

 co" sin 1 



and similarly, 



t (n't + e) = - 3l"-43 sin X + 3"-04 cos X, 



o2 



