OF THE TIDES IN THE PORT OF LIVERPOOL. 
11 
we make c =s siny, we have for the maximum tan 2 {& — l!) = tan y ; and therefore 
maximum 2 — l!) = y. Hence we have y for each H. P. by reducing the above 
double maxima to arcs, and thence we have c, by finding the sines of these arcs. 
16. By the equilibrium-theory, c should be inversely as the cube of the H. P. ; 
c 5 *7^ 
therefore, if C be the value of c for H. P. 57', we have qj = ; logc + 3lo g p 
= log C + 3 log 5 7 ; and therefore this quantity, log c + 3 log p, should be constant. 
It is found that we get a quantity much more nearly constant by taking log c + 22 
log p. The following is the result : 
H. P. 
y. 
log c 
(c = sin y). 
logp. 
22 
10 lo S P- 
log c p 2'2. 
54. 
O / 
25 15 
9-62999 
1-73239 
346478 
3-46478 
3-44125 
55. 
24 15 
9-61354 
1-74036 
348072 
3-48072 
3-44233 
56. 
23 12 
9-59543 
1-74819 
349638 
3-49638 
3-44145 
57. 
22 15 
9-57824 
1-75587 
351174 
3-51174 
3-44115 
58. 
21 18 
9-56021 
1-76343 
352686 
3-52686 
3-43976 
59. 
20 10 
9-53751 
1-77085 
354170 
3-54170 
3-43338 
60. 
61. 
19 45 
9-52881 
1-77815 
355630 
3-55630 
3-44074 
Hence, c = C J nearly. 
1 7- We may, however, express the result more conveniently for some purposes by 
expanding this expression ; for the variation of the maximum will be very nearly as 
the variation of the parallax ; and the double maximum may be nearly expressed by 
the following formula : 
ggm _ 4 m (p _ 57 ). 
The accordance is shown in the lowest line of the second Table in Art. 15. 
18. By comparing, in Table VII. (a), the inequalities for moon’s transit 0 h 30 m , 
l h 30 m , and for 6 h 30 m and 7 h 30 m , it is clear that they are equal to 0 at a later hour 
for the larger than for the smaller parallaxes, which also appears by the maxima. 
Hence a is larger for large parallaxes than for small ones. The exactness of the ob- 
servations hardly allows us to determine its variation exactly. It appears, however, 
that it may be sufficiently well represented by a — l h 15 m + 2 m- 5 ( p — 57). 
19. The effect of the lunar parallax on the heights will be found from Table VII. (b.) 
in the same way as the effect on the times, by taking the mean of each column as the 
non-periodical, and the remainder as the periodical, part of the inequality. The origin 
of the measurements is arbitrary, the low water not being given. The non-periodical 
part is represented with great accuracy (except for the extreme parallaxes) by the 
formula 15'22 + *4 (p — 57). The accordance is as follows : 
c 2 
