48 REPORT—1885. 
Applying (17) with X=0, X’=1', a=), and taking the lower sign, 
and changing the sign of the whole, because of the initial negative sign, 
2 
= 0088 pf (tan 2(-—p 1008242) 
in : ; 
0 ee 2b-n)| - (21) 
The sum of N and L. 
In order to fuse these terms an approximation will be adopted. The 
L tide is just as much faster than M, as N is slower, but the N tide should 
be nearly 7 times as great as the L tide; hence the tan2(A—}) in (21) 
will be put equal to tan2(#—v). We then have 
(N)+(L)= = [ (a+ tan 2(u-- v) 1’) (Ncos2(y— v) — Leos2(y—X)) 
+II’(Nsec2(n—1) + Lsec2(\—p)) sin 2¥—») | 
But 
Neos2(W—v)—Lcos 2(p —d) = cos 24,( N cos 2v — Lcos 2X) 
+ sin 2W(Nsin 2r—Lsin2a), 
Then writing 
Nsin2v—Lsin2X (22) 
WN cos2v —Lcos2X 
tan Ze= 
so that « is nearly equal to v, we have 
= " vy— s2 
(3) 4 (Ly SOeS Neos2r— Heos2Y (a1 + tam2(u—s)W)e082(4-—<) | 
: / 
cos 2e 
cos?A - : 
_ cost [ (av seo2(qe—») + bsee2(A—p)) sin 2 — | (23) 
In the symmetrical term for the sun, with approximation as in (16), 
we get 
(T) +(R)=(L—R)Ueos2(p,—-0) - ss ss (24) 
This terminates the semidiurnal tides which we are considering ; but 
before proceeding to collect the results some further transformations must 
be exhibited. 
Let us consider the function D+aD’, where « is small. From (12) 
we see that 
cos?e—cos?A  , 2sindcosé dé 
D 2D! = << ———— +5’ ——- 
= sin?A +2? Tama dt 
Hence, if d’ be the moon’s declination at a time earlier than the time of 
observation by «/2c, then ; 
D+eD'!= 
cos?0’—cos?A 
sin? 
Hence, in (17), 
D+-tan 2(« pr acoso’ — cos? 
+r tan (k—p) 95 Soe . . . . (25) 
when 6! isthe moon’s declination at time .f—57°°3 tan 2(k—p)/2c. The 
period 57°°3 tan 2(k— p)/20 may be called ‘the age of the declinational 
inequality.’ 
