18 



THE ORBIT OF URANUS. 



We then have, considering only terms of the first order with respect to the dis- 

 turbing forces, 



xv T> m'ihn xr 



V t R= -smJV, 17 , 



! k r 



r, m'lhv T.T 



t R = cosJv; 



dR 



, a t T 



= m - cos N; 



where we put for brevity, 



fri -f- in 



Let us represent by $ the terms in Q which are of the first order with respect 

 to the disturbing forces, so that we have 



The general term in R will then give rise in Q to the term 



*V 



~d^ J 



cos N. 



In the case of the action of an outer on an inner planet this expression becomes 



m! / 



m ( n - -L , on 

 ( 2vA -\ ---- 

 a, \ r dv 



xr 

 cos N; 



while in the contrary case it is 



both derivatives being taken with respect to the logarithm of the mean distance 

 of the inner planet. 



In the integration it will be more convenient to substitute for % and /I the mean 

 longitudes counted from the perihelion of the disturbed planet. If we put 



the angle N will become, 



Since corresponding to each set of values of i' and i there are several values of/' and 

 y, it will be convenient in the numerical computation to combine these different 

 terms into a single one, because after forming the derivatives of R there is no need 

 that o, a' and the other elements should appear in an analytical form. If we put 



