270 



(P-l) cos(0 3 -0 2 )-^. P be called R; then 



(cos (0 a -tir a 



+ sin (0 2 -O (1-f e 2 cos (0 2 -O)R 

 , cos (0 2 -*O)Q 



-COS (0 a BT a ) (1 + * 2 COS (0,-flfJ) R 



=2 Vf: {(l+* 2 cos(0 2 -<))Q+. 2 sm(0 2 -<)R 



From these formulae, the secular variations of the elements are 

 obtained without difficulty ; and a new method of integrating the 

 equations for the variations of the eccentricity and longitude of 

 perihelion is given. 



The author then enters upon a minute examination of the mathe- 

 matical character of secular variations, and their bearing upon the 

 methods of approximation to which the problem of three bodies 

 has given rise. It is pointed out that the disturbance finally effected 

 through the medium of a secular variation is not of the order of the 

 disturbing force, or rather of the ratio of the disturbing force to the 

 central force ; but that it may remain precisely the same, though this 

 ratio should be diminished or increased without limit. The differ- 

 ence affects not the aggregate amount of deviation or disturbance 

 caused, but the time in which this aggregate amount is produced. 

 If we consider the undisturbed problem of two planets about a sun 

 as representing motion in two planes inclined to each other at the 

 angle I, and in ellipses having eccentricities e 2 and e a , it is shown 

 that, no matter how small or how large may be the disturbing force 

 produced on each orbit by the other planet, the aggregate amount of 

 disturbance of the planet m z is of the order of the quantities I and e. 2 , 

 and that of the planet m 3 , of the order of I and e 3 . From considera- 

 tions of this nature, which are dwelt upon at length in the memoir, 

 the author concludes that the ordinary direct methods of solution 

 by approximation, being based upon the erroneous assumption that 

 the variations of the coordinates are of the order of the disturbing 



