738 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Let 
Wn — TtTK (tTTTTT - Kl<) 73 K (lx' of' - K K') 
7 ZK=PQ(p‘ l —q~)+2iq{P 2 e x — Q 2 e*) 
7373 kk — (P 2 — Q 2 ) (p 2 —q 2 ) — 2 PQpq(e N +e~ N ) 
zj^vm-KK) =PQ(P 2 -Q 2 )[{p*-q 2 ) 2 - 2 pY]+ P 2 (P 2 -3Q 2 )pq(p 2 -q 2 )e- N 
- Q\3P 2 -Q 2 )pq(p 2 -q 2 )e N - 2 PQp 2 q 2 (P 2 e~ 2N - Q 2 e 2N ) 
zs k (jzr'zs' — k k')— tlie same with. —N' instead of N 
w,=P 2 Q 2 (P 2 -Q 2 )l(p 2 -q 2 y-2p 2 q 2 J+P\P 2 -3Q 2 )y q 2 (p 2 -q 2 ye-^ 
+gi( 3 p2 _ qypY( p*-qzye N -*' + iP 2 QYq\P^~ 2{N ~ Nr) + qw n ~ n '>) 
'/>(N)w,= -^^ 1 {pq(p°-q*)XP i (P !! -m ! ‘-G i (SP*-Qr) + 2P°Q ! ‘(P i -Q i )pY} 
Now 
P\P 2 -3Q 2 y-Q\3P 2 -Q 2 y 2 =(P 2 -Q 2 )(P i +Q i -6P 2 Q 2 ) 
Put therefore 
©=i(P 3 -Q 2 ){ ( ^-/) 3 ( P 4 +Q 4 - 6 P 3 ^) + 8 P 3 QW) • . • (53) 
and we have 
(f>(N } e)W 3 =2(!5G sin g siny.(54) 
Fast diurnal terms (?i-j- 242). 
W 3 =G 2 [ctk W% 2(e e) ° 2 -f- 73 P 73 ' k'^s 2 ^ 6)+g =].(55) 
By an analogy similar to that by which the fast semi-diurnal was derived from the 
slow, we have 
= 1[4 P 2 QY - Q%3P 2 -Q 2 )Y{ p 2 -3q 2 ) -1 SP 2 Q 2 (P 2 - QJp 2 q\ p 2 -q 2 ) 
~P\P 2 -3Q 2 )Y{3p 2 -q 2 ) - 4P 6 QY] . (56) 
and 
$(N, e)W 3 = — 2(*3ioGo sin g 2 sinj .(57) 
Fortnightly terms (2/2). 
W 0 ==|[(i-2^^)(i-2 CT ^VV)+%V w V 2 e -^^- 2h +H OT V ? W''- E > +2h ] . (58) 
