846 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Now let 
E!=-Ie(l-¥e 3 ); E 3 =l-^e= + We‘; E 3 =|e(l - We 2 ); E 1= W 
J 0 =l-~§e 2 +fe*; J,=|e(l-W 2 ); L=fe 2 
(278) 
And we have 
whence 
<E>(a) = E 1 COS (/2P«Pct)PE 3 cos (2/2Pa)pE 3 cos (3/2 + a —cr) 
—(-E 4 cos (4/2pot— 2 ct) 
J 0 + 2J 1 COS (/2 — ot) + 2J 3 cos 2 (fl — rrr) 
St r (a) = J 0 COS a +Jj[ cos (/2pa— rpP cos (fl—a — ct)] 
p J 3 [ cos (2/2 + a — 2zrr)+ cos (2/2 — a — 2or)] 
> (279) 
These three expressions are parts of infinite series which only go as far as terms in 
e 3 , but the terms of the orders e° and e have their coefficients developed as far as e 4 
and e 3 respectively. 
Then substituting from (279) for <E>, Ny and R their values in the expressions (265), 
we find 
X 3 — Y 3 = P 4 [E 1 cos (2y — fl — ut) 4-E 3 cos (2y — 2/2) + Eg COS (2y—3/2 + T5-) 
PE 4 COS (2y—4/2 + 2nr)] 
P2P 3 $ 3 [J 0 cos 2yP J 1 {cos (2y —/2 Pct)P cos (2yP/2 —ct)} 
p J 3 {cos (2y— 2/2p2(zr)p cos (2yp2/2— 2 To )}] 
p ^ 4 [E : cos (2yp/2 p tp p E 3 cos (2y p 2/2) p E 3 cos (2y p 3/2 — tp 
p E 4 COS (2y P 4/2 — 2«p] 
— 2XY = The same, with sines for cosines 
YZ=The same as X 3 -Y 3 , but with -P'Q for P\ PQ(P~~Q' 2 ) for 
2 P~Q' 2 , PQ ?J for O 4 and with y for 2y 
XZ=The same as the last, but with sines for cosines 
A(X 3 pY 3 -2Z 3 ) = l(P 4 -4P 3 (? 3 p^ 4 )[J 0 p2J 1 cos (/2 — ttt) p2J 2 cos 2(/2 —zw)] 
P2P 2 ^ 3 [E 1 cos (/2Pzpp.E 3 cos 2/2 P E 3 cos (3/2 —w) 
PE 4 cos (4/2 — 2tp] J 
Then if we regard — as constant, and remember that y=n£, and that /2 stands for 
/22pe, and if we look through the above functions we see that there are trigonometrical 
