OF THE TIDES IN THE PORT OF LIVERPOOL. 
7 
same at two places ; and since the empirical formulae have not been determined for 
anv places except those of London and Liverpool, we have not the means of disco- 
vering the relation of the constants at various places. The following comparison of 
the data of observation at London and at Liverpool is instructive as far as it goes. 
The greatest difference arising from the mean semimenstrual inequality is the same 
at the two ports, being 88 m at both. This coincidence is striking ; yet I am disposed 
to believe it accidental, although, according to theory, this quantity ought to be the 
same at all places, since it depends only upon the mean ratio of the solar and lunar 
tidal forces ; for the semimenstrual inequalities at different places differ so much by 
other observations, (varying from 79 m at Brest to 96 m at Plymouth,) that I do not 
conceive the difference can arise from the incompleteness of the observations. 
The effects of the parallax and declination at London were given by approximate 
formulae, less exact than those which we have now obtained for Liverpool ; but, com- 
paring the London formulae with corresponding approximations at Liverpool, we 
have the following results. 
If A' represent the value of X 1 for the mean parallax 57' and the declination 0°, we 
have for the parallax p , and declination S, 
at London x! = A' — 3 m ( p — 57) — 132 m sin 2 3, 
at Liverpool X 1 = A' — 2^ m (p — 57) — 84 m sin 2 &, by Art. 15 and 21. 
Also the maximum semimenstrual inequality, 
at London = 40 m + 3 m (p — 57) + 84 m sin 2 &, 
at Liverpool = 41 m + 2 m (p — 57) 4 * 30 m sin 2 S, by Art. 17 and 23. 
Also if H' be the value of h! for the mean parallax and the declination 0°, we have, 
at London h' = H 7 + 0 ft T7 ( p — 57) — 3 ft sin 2 &, 
at Liverpool h! = IT -f- l ft, 47 (p — 57) — 6 ft sin 2 &, by Art. 19 and 24. 
And at London h = l ft '7- 
at Liverpool h = 2 ft, 8. 
12. The resemblances of the formulae at the two places are remarkable, but the 
differences are still more so. The differences in the heights of the tide at different 
places are indeed what we know to prevail universally, and to depend upon local cir- 
cumstances in an intelligible manner : but the differences in time are more difficult 
to explain, since both the tides come from the same origin. The difference in the 
effect of parallax may indeed be due to the inaccuracy of the data, but it is scarcely 
possible that this can be true of the difference in the effect of declination, which 
appears to be in the ratio of 132 to 84 for the non-periodical, and 84 to 30 for the 
periodical, part. Similar discussions of observations at other places will best throw 
light on this difficulty. 
I now proceed to state the method by which the above empirical formulae have 
been obtained. 
