772 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
Hence 
W y ™ =iG 1 {w cos [(x-X , ) + 2(c'-e)-2 
+ij cos [(x~ x)+ 2 ( e '- e ) — —ty) + (N— x}j)— gj 
+V cos [(x'~X') + 2 ( e '— e ) ~ 2 (f — t) - {&'■~ f) ~ gi] 
+i? v cos[(x—x / ) + 2(e'—e) — 2(x/j'— xfi) + (lV— N') — (xJj— i/»') —gjl . (130) 
Sidereal diurnal term. 
From (38) we have 
/ / 
W 2 ,-=G[e 57 c(t 3 , ct— kk)ttt' k (tz'zs' K f K)e~ e -\- t 3 k(tztz — Kk)ts'k(iS 777' — KK)eS] 
To the indices of the exponentials must be added dz(x — X )• ^ ^ ma y be t rea f ec l 
as unity. Hence the expression becomes G [_KKe x ~ x ’~ s -\-KKe~ (x ~ x!)+s '] and 
W 2 /y=iG {ii' cos (x-x'-g) 
+ ij cos [(x-x )-(^-^)-g] 
+y' cos [(x—x)+ (N'— VO — g] 
+i) V cos[(x-x , )-( iVr -^ / ) + (^-f)-g]} • • • ( 131 ) 
Permanent term. 
From (39) we have 
W 0 j~ = f (£— 2 OTOTif k)(^— 2mm kk) 
= jr — kk—kk to our degree of approximation. 
Now 
kk= T(-r -\-f -f ij(e N ~' tlJ ce~ {N ~' P) )) = \{y +f + 2 ij cos (iV— tft )) 
Hence 
w 0 /y — i(^+i' + 2i)' cos (iV— ^f))— i(i*+f z +2i'f cos (N' — xf>')) . (132) 
W 2 and W 0 are the only terms in W which can contribute anything to the secular 
inequalities, unless Diana and the satellite are identical; for all the other terms involve 
e— e, and will therefore he periodic however differentiated, unless e=e'. 
We now have to differentiate W with respect to i', xjj', j', e', JV. The results 
will then have to be applied in the following cases. 
