734 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
As shown above, however, we need here only deal with a single satellite, so that Diana 
and the moon may be considered as identical and the accents may be dropped to all 
the symbols, except in the differential coefficients of W. Also we need only maintain 
the distinction between Diana and the moon as regards JV, N' and e, e ; and after the 
differentiations of W these distinctions must also be dropped. Hence rx only differs 
from A, k from k , zx from —and k from k in the accentuation of N. 
Also since 0'=f2't-\-e', therefore we may replace 6' — 6 in the three 
expressions (37-9) by e'—e. 
If we put sinp = 2pq, tan \j—q/p, and write </>(W, e) for the operation — 
putting N—N', e = e after differentiation; then from (13) we have 
2 pq dN'~p de' 
Also for brevity, let ( f ) (N)=~ (/.(e) =| ; so that </>(W, e) = (/>(A r ) + </.(e). 
The terms corresponding to the tides of the seven speeds will now be taken 
separately, the coefficients in tx, k will be developed, and the terms involving N '— N 
selected, the operation </>(iV, e) performed, and then N' put equal to N, and e to e. 
For the sake of brevity the coefficient r 3 /g will be dropped and will be added in the 
final result. The component parts of W taken from the equations (37-9) will be 
indicated as W l3 W Ib W m for the slow, sidereal, and fast semi-diurnal parts ; as W l3 
W 2 , W 3 for the slow, sidereal, and fast diurnal parts; and as W 0 for the fortnightly 
part. 
Slow semi-diurnal terms (2n—2/2). 
Wj = IFjOW- e) ~ 2fi + srhff %- 2 <* - e) + 2f ‘] . . 
Let 
Since 
Therefore 
xs—Pp—qQeP 
(40) 
OT 4 =P 1 p 4 — &P 3 Qp B qe* + 6P-Q°p‘ 2 q 2 e 2N — &PQ 3 pq 3 e 3N -\-Q A p^ N 
the same with —N' in place of N 
Therefore 
Wl= ±{pyjr 1 6P*Qyp2e N -x'-\-36P i Q 4 pYe Z{N - N ' ) + 1 QP^Yq^’*' 1 
p_Qy e w-x'n je w-t)-2U 
Therefore 
Wj = %A U P 8 Zn Q~ ,l p 8 2n q~ n e 
«(A r -A")+2(e'-e)-2f, 
where n— 0, 1, 2, 3, 4. 
