OF THE TIDES IN THE PORT OF LIVERPOOL. 
5 
proportional to the square of the declination. Therefore the second of the above 
three propositions is established. 
According - as the moon’s parallax is less, and according- as her declination is greater, 
the moon’s tidal force is less, and h ! in the above formulae is less. Yet it is remark- 
able that these two circumstances affect the magnitude of the retroposition of the 
tide in opposite ways. In one case l! is augmented, in the other case it is diminished. 
When the moon’s force decreases, by her receding further from the earth, the tide 
follows the moon at a greater interval; the mean interval increasing from 10 h 55 m 
to ll h 12 m , while the parallax diminishes from 61' to 54'. But when the moon moves 
away from the equator, which also diminishes her tidal force, the tide follows her 
more closely, the interval decreasing from ll h 12 m to 10 h 55 m , while the declination 
increases from 0° to 27°. 
The Liverpool tide happens about 1 1 hours after the next preceding transit ; and 
as the retroposited tide happens about 37|- hours before this transit, we must sup- 
pose the Liverpool tide to be produced at an interval of 48^ hours preceding the 
time at which it is observed, in order to make it agree nearly with the equilibrium- 
theory ; and we may suppose this time to be employed in the transmission of the 
tide along its channel. If we suppose the original tide to lag behind the position 
of equilibrium, we may suppose the amount by which it lags to vary with the 
changes of the moon’s force, to the amount above stated as the variation of X'. On 
this supposition we may suppose the time of transmission of the tide along its chan- 
nel to be constant. Or we may suppose that the changes of the moon’s force not 
only affect the lagging of the original tide behind the equilibrium position, but also 
affect the velocity of transmission to Liverpool. In either of these ways the circum- 
stances of the tide may be hypothetically represented ; but it will, of course, be un- 
derstood that we use such hypotheses at present only for the sake of connecting and 
representing the facts. 
9. The effect which changes in the moon’s force produce upon the retroposition of 
the tide in time, that is, on the value of a in the formulae (a.) and ( b .), is more difficult 
to determine with any precision. It is, however, manifest from the general course of 
the quantities in the Tables, that a is greater as the moon’s parallax is greater, and as 
her declination is greater. This is proved by each Table independently. Thus I have 
collected as the amount of this change, 
2 m 5 ( p — 57') from the effect of parallax on the times (Art. 18.), 
4 m ( p — 57') from the effect of parallax on the heights (Art. 20.), 
75 ra sin 2 from the effect of declination on the times (Art. 23.) ; 
the effect of declination on the heights offers no clear evidence of a change in a. 
Since, in the change of parallax from 54' to 61', the value of a, as given by the 
times, changes from about l h 8 m to l h 24 ra , the retroposition of the tide in time 
varies from about 34 to 42 hours. 
10. The circumstances of the Liverpool tides may be represented hypothetically 
