Q 
THE REV. W. WHEWELL ON THE EMPIRICAL LAWS 
employed by the calculators of Tide Tables) have never been published, as far as I am 
aware, except in the memoir already mentioned. It was therefore a matter of great 
interest to examine whether the formulae obtained from the London tides are con- 
firmed by those of Liverpool, and whether any further light is thrown upon the sub- 
ject by this addition to our materials. 
2. The results of this examination have been very satisfactory. The Liverpool ob- 
servations have both confirmed, in general, my formulae, and have given me the 
means of very much improving them. The corrections for lunar parallax and decli- 
nation, which, as far as they depended on the former investigation, might be consi- 
dered as in some measure doubtful, and probably only locally applicable, have been 
so fully verified as to their general form, that I do not conceive any doubt now re- 
mains on that subject ; and the nature of the local differences in the constants of the 
formula has also in part come into view. This investigation shows, that notwith- 
standing the great irregularities to which the tides are subject, the results of the 
means of large masses of good observations agree with the formulae with a precision 
not far below that of other astronomical phenomena ; as, for example, a fraction of a 
minute in the times, and a fraction of an inch in the heights. 
This precision is the more worthy notice, because the formulae which we obtain 
point directly to a very simple general law of the tides ; namely, that the tide at any 
place occurs in the same way as if the ocean imitated the form of equilibrium cor- 
responding to a certain antecedent time. This Equilibrium-Theory (the constant 
quantities which it introduces being suitably modified,) expresses, with very remark- 
able exactness, most of the circumstances in my results : I will therefore, before 
stating them, explain it a little further. 
3. The theoretical formula for the position of the pole of the equilibrium-spheroid is 
tan 2 0' = 
h sin 2 <p 
h' + h cos 2 <p 5 
where h and h! are the elevation of the spheroid due to the sun and the moon respec- 
tively, <p the angular distance of the moon from the sun, 0' the angular distance of 
the pole of the spheroid from the moon’s place. 
In the case of the tides, we may suppose the actual ocean-spheroid to follow the 
equilibrium-spheroid at an angular distance A', the spheroid being that which corre- 
sponds to a distance of the sun and moon <p — a, instead of <p. Thus we have 
tan 2 (Jf — X 1 ) 
h sin 2 (<p — «) 
h 1 + h cos 2 (<p — 
In the same manner the theoretical height of the pole of the equilibrium-spheroid 
above the mean surface is V {A' 2 -f- A 2 + 2 A A' cos 2 <p} ; and on the equilibrium-theory 
the height of the tide above the mean surface is v'{A ,z + A 2 + 2 A h! cos 2 (jp — a)}. 
Uv assuming properly the values of A and A', a and X', these expressions may be made 
to agree very closely with the mean results of observation. This was shown with 
