AND ON THE DIURNAL INEQUALITY OF THE TIDES AT LIVERPOOL. 139 
These differences in the value of a may arise from the lunar declination not being 
completely eliminated in our previous calculation, the interpolation of the Table 
being slightly inexact. But it is by no means improbable that they are the discrepan- 
cies, arising from mechanical principles, which exist between the tides of water in 
motion and the results of the equilibrium-hypothesis. The general agreement of the 
equilibrium-theory with the facts (assuming « = 2 h ) is near enough to be very re- 
markable. It may be possible, by additional care and labour, to bring the solar inter- 
polated curves nearer to the observations, preserving the requisite conditions : but I 
conceive that enough has been done to establish the general law. 
§ 4. On the Solar Inequality of the Time of High Water at Liverpool. 
18. The solar inequality of the time must be found in exactly the same manner as 
the solar inequality of the heights. By interpolation of Mr. Lubbock’s Table XVI., 
I have obtained Declination Table (W. T.) ; and by comparing the time of high water 
due to lunar declination by this Table with the observed times as given in column 
(L. II.) of the calculations, I obtain the residual quantities in the columns (DifF. T.), 
which should exhibit the solar correction of the times. 
I then find the mean of the column for each month, and subtract it from every 
number in the column, in order to separate the non-periodical from the periodical part 
of the residual quantities. The means and the remainders are exhibited in Table 
(B. T.). The remainders were laid down by coordinates, but I have not thought it 
necessary to give these curves. 
The points thus found being joined by continuous lines, I had a series of curves 
which ought to exhibit the solar inequality. These lines were less obviously regular 
than those which we obtained by a similar treatment of the heights ; but they were 
still apparently free from any lunar effect, which would have given a maximum passing 
successively from one month to another ; and I could trace a solar cycle in them ; 
namely, the curves for l h 30 m , 2 h 30 m , and 3 h 30 m had a maximum about April, and a 
minimum about August, while the curves for 5 h 30 m , 6 h 30 m , 7 h 30 m had these features 
inverted, and the rest of the curves had small ordinates only. 
19. Let us compare the laws of the phenomena of which we thus catch a glimpse 
with the laws according to the equilibrium-theory. We have, on that hypothesis, 
tan {& — X') = 
k sin 2 (<p — «) 
k' + h cos 2 (<p — «) 
= t suppose, 
where 6 ] is the interval of the tide and moon’s transit. 
Now let h become h + A h, and & become + A we have, approximately, 
tan (f — X') -j- 
d . tan (6' — x!) 
dUf 
A^t + ^Ah 
sec 2 (0 = p k Ah 
