SECULAR VARIATIONS OF THE ELEMENTS OF 



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



h=N sin (#<+/?), h'=N' sin (grt+0) h"=N" sin (gt-\-(3\ &c, ) g > 

 l=N cos (^-|-/3), Z'=tf' cos \gt-\-p) l"=N" cos (^+/3), &c - * 

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

 Ng ={(o,i)+(o,i)+(o,»)+ &c.| N—l^T\N'—\^J\N"—[^]N'"— &c. *] 

 jy , flr = |(i,o)-i-(i,2)+(i,3)-i-&c.| N'—Fr^}N—[T^}N"—[T-P\N'"—&c. I ^ 

 ^"y=j(a,o)-|-(i,i)-|-(i,«)+ &c.| iV"— I^W— [iTTIlV' — [^J|iV w — &c. I 

 &c. "' '"" ~ ~ J 



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



3. The quantities (o,i) and (i,o), [oT], [TJ\ ; (0,2) and (2,0), [oTT], {TJ\; (1,2) and 

 (2,1); [T7i] and |TT| , &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 



Sin' no? a' (a, a')' 



(o.O = 



(4) 



4(a' 2 — a 2 ) 2 



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 n', a', and rn', a' into m, a, respectively, (0,1) will change it into (1.0), and 

 we shall have 



3mn'a' 2 a(a\a)' 



(1,0) 



(5) 



4(a' 2 — a 2 ) 2 

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



we shall also have 



0. °)=(°. 0- 



n 



(2,0) =(0,2) 



(3,0) = (0,3) 

 &c. 



ra'es' . 



(6) 



na 

 m 



na 



n a" 



ni 

 n'"a" 



rn'" 



(7) 



The same relations also hold with respect to the quantities |o,i| , |i,o| , |o,2| , |2,o| , 

 &c, so that we shall also have 



ni n'a' 

 na rn' 

 v.- n"a" 

 na m" 



[TTo]=[o7T]. 



r^i^r^ 2 !- 



[H o ]=[EI] 



m 



na 



&c. 



(8) 



