SECULAR VARIATIONS OF THE ELEMENTS OF 



2. Now to find the integrals of equations (A), we shall suppose 



7t=iV sin (flr+/3), h'=N' sin (gt+Q) h"=N" sin (/<+/?), &c., ) g 

 I=N cos (0<-j-0) ?=#' cos (gt+P) l"=N" cos (flrt-j-0), &c. ) ^ 

 If these values be substituted in equations (A), they will become, 



Ng ={(o,i)+(o,z)-j-(o,s)-f- &c.\ N [oTHiV [oTTj^V" [oT^Y'" &c. 







/ 27^"'" &c. 



&c. JV> To^ 



&c. 



If we suppose the number of planets whose mutual action is considered, to be i, 

 the number of these equations will be i; and by eliminating the constant quantities 

 N, N 1 , N" t &c., we shall obtain a final equation in g of the degree i. 



3. The quantities (0,1) and (1,0), [0,1 1, [ I,Q| ; (0,2) and (2,0), |o,2|, 1 2 . o | ; (1,2) and 

 (2,1); |i,8| and |2,i|, &c., have some remarkable relations with each other, which 

 not only facilitate their computation, but render the equations resulting from the 

 elimination of N, N', N", &c., much shorter and more commodious. The general 

 expression for (0,1) is 



In this equation n and a denote the mean motion and mean distance of Mercury, 

 m' denotes the mass of Venus and a' its mean distance from the sun. If we change 

 n, a into ', a', and m', a' into m, a, respectively, (0,1) will change it into (1,0), and 

 we shall have 



Now since (a, a')'=(a', a)', equations (4) and (5) will give 



m n'a' 



we shall also have 



&c. 



na m' 



na m 



na m" 



(6) 



The same relations also hold with respect to the quantities [oTT], [To], [oTFI, [TTo], 

 &c., so that we shall also have 



rri=|TT|~ n ' a> 



8 , = I , i 



3,0 = 0,3 



na m' 



na m" 



na m'" 



&c. 



(8) 



