OF THE TIDES IN THE PORT OF LIVERPOOL. 
3 
respect to the expression for the time by Mr. Lubbock from the London observations. 
The Liverpool observations give a still closer agreement, assuming k' = ] l h 6 m , 
a = l h 15 m , y = sin 89 m = sin 22° 15'. 
The expression for the heights also agrees very nearly with observation, as I shall 
show, but for this purpose we must suppose a = l h , = sin 23° 30'. 
The agreement in these cases is the more remarkable, on account of the want of 
symmetry in the functions which thus occur. The curve, the ordinate of which re- 
presents the time of high water (reckoned from the moon’s transit), is not symme- 
trical with regard to its maximum ordinates. The curve, the ordinate of which 
represents the height of high water, is not symmetrical with regard to its mean line 
of abscissas. Yet in both these cases the theoretical and observed curve agree within 
a minute and an inch during their whole course. It is impossible to doubt, under 
these circumstances, that the theoretical formula truly represents the observed facts. 
4. But this agreement belongs to the mean of all the observations ; and we have 
further to seek for the alteration in the formula, which is requisite in order to repre- 
sent the effect of changes in the parallax and declination of the sun and moon. In 
these respects also we find a near agreement of the theory and observation. By the 
equilibrium-theory, the height h! of the lunar tide ought to be proportional to the 
cube of the moon’s parallax ; it is exactly or nearly so : by the same theory the 
height h! ought to diminish when the moon’s declination increases, and by a quantity 
proportional to the square of the sine of the declination. It is found to do so with 
great precision. 
5. But the equilibrium-theory, since it does not point out the existence of the quan- 
tities k' and os, does not indicate what changes these quantities may be expected to 
undergo, when the moon’s force is altered by the effects bf parallax and declination. 
We find that in that case, these quantities also are altered, and the resulting change 
in the phenomena may be conceived in the following manner. 
If we suppose the moon to revolve about the earth by the diurnal motion, perpe- 
tually drawing the waters towards the position of equilibrium, we may conceive that 
the ocean would form a spheroid, the pole of which would revolve round the earth, 
following the moon at a certain distance of terrestrial longitude. For the sake of 
distinctness, let this distance be called the Retroposition of the theoretical tide in lon- 
gitude. Its mean value is what I have termed in other communications the “ cor- 
rected establishment" of a place in the open ocean. 
If, from an original equilibrium-tide, a derivative tide were sent off, along any 
channel in which it is no further influenced by the forces of the moon and sun, it 
would take a certain time in reaching any place in that channel ; and the circum- 
stances of the tide at that place would not depend upon the positions and distances 
of the moon and sun at the time when the tide happens, but upon the positions and 
b 2 
