AND ON THE DIURNAL INEQUALITY OF THE TIDES AT LIVERPOOL. 135 
bock’s Tables I., II., III.* In these tables the parallax may be already considered as 
reduced to its mean value ; for each number is the result of about ninety observations, 
distributed through nineteen years, in which time the moon’s perigee moves twice 
round the ecliptic ; and hence the deviations above and below the mean will balance 
each other. But it is otherwise with the declination ; for since the mean moon may 
be considered as moving in the mean elliptic, a certain time of the year, with a certain 
hour of moon’s transit, implies a certain declination ; for the moon’s time of transit 
added to the sun’s right ascension is the moon’s right ascension, which of course de- 
termines the declination. Hence the variations of the numbers in Tables II. and III. 
are those which arise from the varying forces of the mean moon and of the sun ; and 
the greater part of the variation arises from the change of the declination of the moon. 
This part must be eliminated, in order to bring into view the solar effect ; and we are 
able to perform this elimination by means of Table I., since that Table contains also 
the mean declination of the moon for each set of observations. In each set, the num- 
ber of observations being large, and the limits of declination small, the mean correc- 
tion for the lunar declination may be taken to be identical with the correction for the 
mean lunar declination, although the correction varies nearly as the square of the 
declination. But in order to apply this correction, we must have a table of the effect 
of every degree of declination ; whereas Mr. Lubbock’s Tables XII. and XVI. only 
contain the effect for every 3° of declination. It is obvious also, by inspection of this 
table, that it is affected by many casual irregularities which must be got rid of by 
interpolation. Among these irregularities, however, I do not include the apparent 
difference of the effect of north and south declination, which is shown in Tables XI. 
and XV. ; since it will be seen by comparison with Table X. that the greater part of 
this difference arises from the effect of parallax ; for the changes of parallax and de- 
clination so nearly recur in the same cycle, that their effects are not insulated in a 
period of nineteen years. The remainder of the difference arises from the solar in- 
equality of which we are in search. 
12. I have therefore laid down Mr. Lubbock’s Tables XII. and XVI., and interpo- 
lated them by means of curves, and have thus obtained two new declination tables 
for the Times and the Heights. These tables I shall designate as Declination Tables 
(W. T.) and (W. H.) respectively ; meaning by the former letter to distinguish them 
from Mr. Lubbock’s Tables XII. and XVI. , They may be used in calculating the 
time and height of high water at Liverpool (the table for the height being altered by 
a constant if the measures are not taken from Mr. Hutchinson’s zero). When so 
applied, they take the place of Mr. Lubbock’s Tables XXV. and XXVI., from which 
they differ — in including the semimenstrual inequality, in being interpolated indepen- 
dently, in being given for the middle of each hour instead of the beginning, and for 
every degree of declination instead of every three degrees. These Tables will be found 
at the end of this Memoir, namely, 
* Philosophical Transactions, 1835, p. 282. 
