46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 361 



element for any given time by a simple integration. But this is not the 

 case. 



The method of variation of elements gives us, not a solution, but merely 

 a transformation of our original differential equations of motion. The rates 

 of variation are given in terms of the unknown elements themselves ; and in 

 order to find the elements from the equations so formed, we must employ 

 repeated approximations. 



Let us consider this matter a little more particularly. 



The terms which express the rate of variation of any element may be 

 divided into two classes: 



1. Those which involve the mean longitudes of one or both of the 



planets concerned, as well as the elements of their orbits. 



2. Those which involve the elements only. 



The first are called periodic terms, since they pass from positive to 

 negative, and vice versa, in periods comparable with those of the planets 

 themselves. 



The second are called secular terms, and vary very slowly, since the 

 elements on which they depend do so. 



Each of the terms in the expression of the rate of variation of any 

 element will involve the mass of one of the disturbing bodies as a factor. 



Hence, if all these masses be very small, all the periodic inequalities of 

 the elements will be likewise very small, and we shall obtain a value of the 

 rate of variation which is very near the truth if we substitute for the com- 

 plete value of any element its value when cleared of periodic inequalities. 



Then the periodic inequalities in the element under consideration may 

 be found by direct integration, supposing the elements to be constant in the 

 terms to be integrated, and the mean longitudes only to vary. 



Also the secular variation of the element considered, that is the rate of 

 variation of the element when cleared of periodic inequalities, will be given 

 by the secular terms taken alone. 



If the disturbing masses, however, are not very small, this process is 

 not sufficiently accurate, and the periodic inequalities thus found can only be 

 regarded as a first approximation to the true values. 



A. 46 



