THE ORBIT OF URANUS. 181 



Differentiating with respect to ^, we find for the terms of the second order 



6-— = — iii'anl\ sin NhN 

 at 



h-^- = — m'anl\ cos Nh N. 

 at 



Substituting for ^iV^ its numerical vahie just given and integrating, we find 



Ih = m'avh \ —2895" sin N— (6G.38" — 4".'2G0 cos N -\- 1815" sin 2X\ 



— 'iQ'iQ")iial\nt -\- constant ; 

 lh = m'avh\ 1—2895" cos iV+ (6658"— 4".260 sin .¥+ 1815" cos 2xV| 



-|- constant; 



the constants being so chosen that hh and hlc shall vanisli at the epoch. 

 Reducing the values of hh and hk to numbers, tliey become 



cVi = 5".82 sin iV+ (13".40 — 0".86 T) cos N— 3".65 sin 2.V+ VmT— 2".67, 

 hh = 5 .82 cos ^Y— (13 .40 — MT) sin N— 3 .65 cos 2.V + 12 .12, 



the last two terms being arbitrary constants. 



Computing the values of these terms of hi, (h, and H', for intervals of 50 years, 

 from 1600 to 2000, we find them to be as follows: 



Year H 



1600 — r'.3-l 



1650 —0 .71 



1700 —0 .31 



1750 —0 .10 



1800 —0 .01 

 1850 .00 



1900 .00 



1950 +0 .04 



2000 +0.18 



We see that althougli the ultimate effect of tliese terms is very considerable, 

 their effect, during the period that Uranus has been observed, is insignificant. 



Concluded Elements and Perturhntions of Vranns. 

 The corrections found in the last chapter being applied to the final provisional 

 elements (p. 99) give the following elements for 1850, affected by the great 

 inequality produced by Neptune: 



Elements IV of Uranus. 



Epoch, 1850, Jan. 0, Greenwich mean noon. 

 Tt, 168^ 15' 6". 7 



e, 28 25 17.05 



0, 73 14 8.0 



,^, 46 20.54 



e, .0469236 



