788 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Kl-fa— —(/Co-f y8)l 
K z J r a= 
( 192 ) 
ab(a+£) = (/c 1 + a)(/e a +»)(*! +#c 2 ).(193) 
Now suppose our equations (181) to be written as follows:— 
dz 
= ay + a^ + S 
— —az —a £ + w 
= firj+by+o- 
dt 
dy 
dt 
d£ 
dt 
(194) 
Where s, u, cr, v comprise all the terms involving a, a', &c., y, g, &c. 
Then if we write (z) as a type of z, y, £, y; (a) as a type of a, a, /3, b; (a') as a type 
of a, a', /3', b'; (y) as a type of y, g, S, d; and (s) as a type of s, u, cr, v; it is clear 
that (s) is (z)(a')t-\-(y)(z). 
Differentiate each of the equations (194), and substitute for after differen¬ 
tiation. Then if we write 
q_ ds . 
— , -f - au -J- an 
dt 
The result is 
<Pz 
dt 
TT du 
U = ~r~ — as — a<r 
dt 
2=^'+/3i> +b u 
T = y-P<r -bs 
dt .j 
2 = ~(a 3 +ab)z —a(a+yS)^-f S 
. (195) 
yp — — (a- +ab)y — a(a-f/3)iyd-U 
d^ 
(19.6) 
y t 2 —— (/3'+ab)^ — b(a-f/3)z-fS 
= —{fi 2 -\-&b)r)—b(a-\-/3)y-] r T 
From the first of these 
d~z 
— (£ 2 +ab)a(a+/3) £= (/3 a d-ab) — + (a 3 +ab)(/3 2 +ab )z — S(/3 2 +ab) 
