70 



THE ORBIT OF URANUS. 



Beginning with the last two terms of this expression, it may be shown at the 

 outset that they are quite insensible. The effect of the constant terms in hrj and 

 fy/ has already been included by correcting the logarithm of the mean distance by 

 their amount ; they are therefore omitted. The largest remaining term is 64", the 

 square of which is only 0".()2. In the product r^p 2 the largest terms are 



+ 0.014 



0.013 sin g 

 -0.011 sin (30 

 -0.011 cos (4^ 



which may be entirely neglected. 



We shall therefore only consider in &Q the terms 



+ 



~- 



dp 



As already remarked, R is rigorously a function only of V, p, and p', V being 

 the angle made by the radii vectores of the two planets. But, in the analytical 

 development of E, the quantity V is considered as a function of v, v', and y, so 

 that we have 



In the previous computation of the perturbations of Uranus, we have supposed R 

 to be a function of p , p' , etc. The corrections to R and its derivatives with respect 

 to v and p are now given by the equations (11), with the modifications shown on 

 pages 24 to 27. The derivatives of R which enter into these equations are formed 

 as follows: If, in the value of R produced by the action of Saturn on Uranus, we 

 consider any term of the form 



m'h 



- COS iV 



where 



N= 



the accented quantities always referring to Saturn, but ! being the corrected mean 

 distance of Uranus, then we shall have the following terms in the derivatives of R. 



m'h , 



<9v 



a i 



d* 



d_R 

 dR 



w~ 



Osin N 



in N 



m' , dh \ 



-(/* + -- ) cos N 

 tti \ dr>/ 



m' dh 



cos N 



m'h 



