850 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
<£(x) = i|jP 8 F x sin 2f x fi- 2P 6 Q~G x sin g x — 2 P 2 Q 6 G X sin g x —-|Q 8 F X sin 2 f x ^ 
— 3P 4 (hH s sin (xh x ) ! 
. 7 'r (285) 
i//(x) = 2 P 4 () 4 (F x sin 2 f x — F x sin 2 f x )-}- 2 P 2 $ 2 (P 3 —Q 2 ) 2 (G X sin g x —G x sin g x ) | 
-i(P 4 -4P 2 () 2 +() 4 ) 2 H x sin (xh x ) J 
And the generalised definition of the F’s, G’s, H’s, &c., is contained in the following 
schedule 
speed 
2 n — x/ 2 , 
n— x/ 2 , 
x/ 2 , 
11 + x/ 2 , 
2 ?i+x /2 
height 
F x 
G x 
H x 
Gr x 
F x 
>. . 
. (286) 
lag 
2 f x 
g x 
(xh x ) 
gx 
2 f x 
We must now substitute for the E’s and J’s their values, and as the ellipticity is 
chosen as the variable they must be expressed in terms of r/ instead of e. Also each 
of the E 2 ’s and J 2 ’s must be divided by ( 1 —e 2 ) 6 . 
Then since v 7 1—e 2 =l— r/, therefore 
e 2 = 2r) — yf and ( 1 —e 3 ) '' = ( 1 — -q ) 12 = 1 fi-12^4-78?]' 
E, 2 
(1 — V? 
(1 - 77) 12 
and 
and n Pt,= 289 V 3 
Then by (278) 
Ei : =i e ~(i— SL 4 e ~)=iv(i — 13 y) > - (J l _‘^12=hi 1 —v) 
Ey=l-lle 2 + ^e 4 =l-227 7 + A 4V 5 and ^^=1-107?+++ 
E 3 2 =¥e 2 (l-We 2 ) =W(1 ~Hh) 
E, 2 =HPe 4 =289?r 
J 0 2 = 1 — 3e 2 -f 3e 4 = 1 — 617 +15+ 
Ji 2 =l e 2 (l-¥e 2 )=!^(l- 8 ^) 
J 2 3 =fi e4 =¥^ 3 , —(!_,) 
When y is put equal to - we shall also require the following 
b 2 2 ( 2 ) 
0-—v? 
and c¥ 
a " rl > l ft“ = L | ' l + 4r ') 
; 1 + 677 + 2177 2 
and — 
GHl+v) , 
+l —P 12 25 
E 3 2 (3^-l) 
( 1 — 77 ) 
^= 2(1 — 1077) ; 
77 ( 1 — 77) 12 
J1T77-1: 
77 ( 1 — 77) 12 
J 3 2 ( 2 t 7 — 2 ) 
77 ( 1 — 77) 12 
77 ( 1 — 77) 12 
!> • (287) 
(288) 
1(1 + 07]); 
