736 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Let 
w n = (tzkts'k)' 2 
Tz=Pp — Qqe N , k= Qp-\-Pqe N 
txk=PQ(p 2 — q 2 )-{-pq{ P 2 e~ N — Q z e N ) 
K'K'=PQ(p*-q*)+pq{P*e N '-Q*e- N ') 
^/w u =P 2 Q 2 {p^-q 2 Y+PQpq(p 2 -q 2 )[P 2 {e N '+e- N )-Q\e- N '+^ 
+._py[p%-(*-*') 4 . Q 4 e (M') _ p*Q2( e x+N> _j_ e -cy+ifO)] 
w U — P i Q\_ (pr—cpY — Aqrq~ (pr — q~) ~ + Apficf] + 4 P 6 Q 2 pr(p( pr — <f) ~ e ~ (K ~ N ' ) 
+ AP 2 QYq 2 (p 2 -q 2 Ye N - N ’+pY[_P 8 e- 2{N - N,) + Q 8 e 2{N ~ N '>] 
<f,(N) w u = - 5 ^ 7 = [4ps(^-? 2 ) S ^ i <? S (f 1 -e‘) +W(^ 8 -<? 8 )] 
If we had operated on the other term of T\ T ji we should have got the same with the 
opposite sign, and e~ { in place of e“~ f . 
Then let 
&=±(P*-Q*){2{p*-q*)*P*Q*+py(P*+Q*)} .... (44) 
and we have 
<£(iV, e)W„=2#F sin 2f sin^.(45) 
Fast semi-diurnal terms (2n-\-2f2). 
W nI =iFo[K 4 K /4 e- 2(e '- f) - 2f =+K 4 K% 2(e '- £)+2f =].(46) 
Since k is obtained from to by writing Q for P, and — P for Q, therefore by writing 
— 2b for 2f L , and interchanging Q’s and P’s we may write down the result by symmetry 
with the slow semi-diurnal terms. Then let 
^jr=-f[(? 8 y> 6 — 4 P 2 Q & p\ p 1 — 3q°) — 1 SP i Q i p>°q 2 (p 2 —q 2 ) — AP 6 Q 2 qA(3p 2 —q°) — P s g 6 ] . (47) 
and 
e)W m = — 2db,Fo sin 2b sinp.(48) 
Slow diurnal terms. 
W X =G ^TX ? 'KTZ5 , ''’K e 2(fl e) gl -f- VpKTS^Ke’ 
■2 ( 6 '-u+ gl - 
(49) 
Let Wj^^/fw'h'e 2 *' e) gl . 
