470 Royal Society : — 



e(to + cos . ^*/')=NA¥r^* { ^"^(^-^) Pf.0-(^ +ecos(0-T<r)) P,,„ | , 

 2 



which are capable of being expanded in terms of the elements, and t 

 by means of the ordinary expressions for r, 6, and d—'ur in terms of 

 the same quantities. The values of the elements at the time t being 

 supposed to be found, by the integration of these formulBe, in terms 

 of t, and their initial Aalues, are to be substituted in the ordinary 

 expressions for the coordinates, so as to obtain their values at the 

 time t. 



The author exhibits the application of the preceding formulse to 

 certain simple examples, and then proceeds to apply them to the 

 planetary theory. For two planets (distinguished by the suffixes 2 

 and 3) supposed to move in the same plane, the following are the 

 rigorous expressions for the variations. Let a^ and a, be the ratio 

 of the mass of each planet to that of the central body. Let P denote 

 the cube of r,j-i-r„^, and let (P— 1) sin (0^—d.^) be called Q, and 



(P— 1) cos (03— 0,)—^ P be called R; then 

 ''3 



^^"-= %2^lSe^) i { (^°s (y.-^.K2 + e, cos(9,-z<r,)) + OQ 

 + sin (0,,— tij,,) (l + e.cos (0.—^S))'R\ , 



^ N A >■ f 

 ej^.,=cx,, ; ■' \ , < sin (0,-ra-,) (2 + e, cos id,-^^))q, 



— cos (0,— zs.J (1 4- e.-, cos (d^—m.^)) R i . 

 ^A2=2a3_ZiAL.^ |(i + e^cos(0,-t^,)) Q+e,sin (d,-m.jjx\, 



^(N(< + r)) = - (i - e^)* tV -2«3 ^^4^11. 



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 



