OF THE TIDES IN THE PORT OF LIVERPOOL. 
13 
The first line of that Table contains the non-periodical part. In order to find its 
law, subtract each mean from ll h 12 m corresponding to decl. 0. We obtain a series 
of numbers which increase faster than the declination ; and it is found that they may 
be nearly represented by the expression 84 sin 2 &, c$ being the declination. The agree- 
ment is as follows : 
Declination 
0°. 
3°. 
6°. 
9°. 
12°. 
15°. 
18°. 
21°. 
24°. 
27°. 
Obs.Diff 
0 
1-0 
0-7 
1-7 
3-6 
5-5 
8-0 
10-8 
13-0 
16-4 
Formula .... 
0 
0-2 
0-8 
2-0 
3-6 
5*6 
8-0 
10-8 
13-9 
17-3 
Hence the non-periodical part is ll h 12 m — 84 m sin 2 &. 
The remainder of the Table XII. fa.) exhibits the periodical part of the inequality ; 
and it will be seen that each column follows nearly the law of the mean semimenstrual 
inequality as already obtained. In order to obtain the law of the coefficients, I take, 
as before, the sum of the two maximum values. This sum converted into arc gives 
7, and c = sin 7. 
Decl. 
V- 
1 
cosec y = — . 
r c 
Excess of 
Decl. 0°. 
Log. 
Excess. 
Log. sin 2 2. 
Difference. 
O 
0 
20 33 
2-8488028 
3 
20 36 
2-8421877 
6 
20 54 
2-8031777 
9 
20 51 
2-8091995 
•0396033 
2-59769 
18-38866 
•20903 
12 
20 50 
2-8117471 
•0370557 
2-56878 
18-63576 
1-93302 
15 
21 27 
2-7345630 
•1142398 
1-05778 
18-82600 
•23178 
18 
21 52 
2-6849391 
•1638637 
1-21447 
18-97996 
•23451 
21 
22 23 
2-6260406 
•2227622 
1-34783 
19-00866 
•33917 
24 
23 5 
2-5505680 
•2982348 
1-47455 
19-21862 
•25593 
27 
24 8 
2-4458163 
•4029865 
1-60528 
19-31410 
QO 
l-H 
<Ti 
For the smaller declinations, the differences are too small to be depended on. The 
numbers corresponding to the resulting logarithms from 15° to 27° are from T7 to 2-1. 
If we take the mean T85 as the number, we have for — the value 2'85 — 1*85 sin 2 c>, 
which is sufficiently near. 
1 h' 
22. By the theory of equilibrium — = y. And by the same theory, if IT be the 
height of the lunar tide at the equator when the declination is 0, we shall have in 
latitude l, when the declination is &, two tides, of which the heights are H' cos 2 (Z + l) 
and H' cos 2 (l — l). Now as we have not distinguished these two tides, our result 
will be the mean of them. Therefore, 
h 1 = \ IT {cos 2 (Z + S) + cos 2 (Z — h) } 
= H' { cos 2 1 cos 2 S + sin 2 1 sin 2 & } 
= H' {cos 2 Z — (cos 2 Z — sin 2 Z) sin 2 
= H' cos 2 Z { 1 — (1 — tan 2 Z) sin 2 &}. 
