70 THE ORBIT OF UK A XUS. 



Beginning witli the last two terms of this expression, it may he shown at the 

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

 ^p' has already been included by correcting the logarithm of the mean distance by 

 tlieir amount ; they are tlicrefore omitted. The largest remaining term is 64", the 

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



+ 0.0U 



— 0.013 sin g 



— 0.011 sin (3r/ — 20 



— 0.011 cos (4r/ — 2/) 



Avhich may be entirely neglected. 



We shall therefore only consider in bQ the terms 



2 fu^hRdt + ^4- 

 *^ op 



As already remarked, R is rigorously a function only of T'', p, and p', "T being 



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



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



that we have 



^=/(p,p',v,v',y). 



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

 to be a function of po, p'o, 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 /?„ 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 



nth ,, 



cos N 



where 



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. 



(/+y)sin^ 

 :^ (/'+/) sin ^ 



I 



— (7/, + ^r-lcosi\r 



dh -- 



-— — COS ^V 

 dn 



, — (;4-y)-cosiv 



dv- «, 



