AND ON THE DIURNAL INEQUALITY OF THE TIDES AT LIVERPOOL. 
137 
height of the solar tide is added to that of the lunar in the former case (at spring- 
tides), and subtracted in the latter (at neap-tides). 
15. Thus the general course of the phenomena shows the existence of the solar in- 
equality ; but we shall trace its law more distinctly by taking the suggestions of the 
equilibrium-theory. 
If h, h!, be the height of the solar and lunar tides, the angular distance of the sun 
and moon, y the compound tide, we have 
y = y' {h 2 + h! 2 + 2 h h! cos 2 (<p — a)}, 
as shown on former occasions ; a being a quantity which is to be determined so as to 
accommodate the equilibrium-theory to the actual case. 
Let h undergo any change so as to become h + A h, A h being small. Then y be- 
comes approximately 
y+Ay=y + ^|-AA=y + 
h + fi! cos 2 (<p — «) 
*/ \h 2 + A' 2 + 2 h h! cos 2 (<£ — «) } 
Ah. 
The quantity A h varies according to the different season of the year, and depends 
principally on the sun’s declination, the changes of solar parallax being too small to 
affect much the amount of solar tidal force. The curve which expresses the changes 
of A h has, as appears in the figures, a maximum at the end of March, soon after the 
time when the sun is in the equator, and when, consequently, his tidal force is the 
greatest. It cuts the axis in May, soon after the sun has his mean effective declina- 
tion : it has a minimum in July, soon after his greatest declination, at which time his 
force is least. The mean effect recurs in the end of August : there is another maximum 
in the end of September. About November, December, and January, there is another 
mean and another minimum, arising from the mean declination in November and the 
greatest declination in December. Thus the general course of the value of the cor- 
rection for a given value of <p agrees with the equilibrium-theory. The want of perfect 
regularity in the form of the curves is due partly to the combination of the effects of 
solar parallax and solar declination. According to the theory, the greatest amount 
of the correction for solar declination is about one tenth, and the greatest amount 
of the correction for solar parallax about one twentieth, of the whole solar tide. 
16. For a given season of the year, if we follow the changes of <p through twelve 
hours, we easily see from the formula that Ay has a continuous series of values, 
among which are a maximum and a minimum value. Hence the curves in Plate XVII. 
ought to be such that the ordinates for each month, taken in the successive hour- 
lines, form a continuous series. The subtraction of the non-periodical part, which 
we have performed, will not affect this continuity, since this part is constant for the 
month. 
Hence in correcting the original curves of Plate XVII., so as to get rid of irregula- 
rities, we must endeavour to make them conform to these two conditions : — that the 
ordinates for the same month shall form a continuous series, with a maximum and 
minimum ; and, that the curves for the different hours shall be similar to each other, 
MDCCC XXXVI. T 
