18 THE ORBIT OP UllA NUS. 



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

 turbing forces, 



D,R = sin N, 



«i 



(17) 



D^R^ cos N; 



a, 



1 

 h 



where we put for brevity. 



6R , tti 



^— — «t^r— cos iV; 



n 



in -\- in 

 N = iX + a +/J +/co. 



Let us represent by Q^ the terms in Q which are of the first order with respet-t 

 to the disturbing forces, so that we have 



Q, = 2Jn,R,Jr^^- 



The general term in R will then give rise in §o to the terra 



h 



2ihv a^ 



cosiV. 



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



while in the contrary case it is 



'^(2u7i_h-tK)cosN, 



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 X and "k the mean 

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



^ = i/ + » 

 the angle N will become, 



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

 y, it Avill 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 (J, u' and the other elements should appear in an analytical form. If we put 



