ON THE LAWS OF RISE AND FALL OF THE SURFACE. 
257 
If we take <p' — <p to be 24 minutes, and c to be — , which is the value which agrees 
best with the times at Plymouth, these tidal half-days are respectively 12 h 17 m and 
12 h 40 m . At any other time 
. tfi;.. l+ccosa? r 1 
T — i2 + J4 1 + 2ccos2p + c* — li I 1 +30 
1 + c cos 2 <p 
1+2 c cos 2 <p + c~ 
I 
The height, 2 h, of the total tide varies with the period of the lunation. If 2 H be 
the total lunar tide, 
at springs h = H (1 + c'). 
at neaps h = H (1 — c'). 
at any time A = H\/{l-j-2c' cos 2 -j- c' 2 } ; 
but c' has no longer at all places the same value as c before. The Liverpool tides 
give -r as the value of c but in order to satisfy the Plymouth observations of height, 
c' must be about -j. 
Since the curve of rise and fall at spring tides has its amplitude the smallest and 
its ordinate the largest, it will intersect all the others. The points of intersection 
will be a little above mean water. 
We have now to inquire whether these features appear in the empirical laws of the 
tides, as collected from discussion of the observations already spoken of : namely, 
whether the curve of rise and fall is the figure of sines, and whether its maximum 
ordinate ( h ) and its amplitude (r) follow laws resembling those given by the theory. 
If this is the case, the intersections of the curves will also agree with the theory. 
(1.) The empirical curve of rise and fall agrees very nearly with the figure of sines 
at all the three places here considered. The figure was determined differently at the 
different places. At Liverpool, where the height of the water was observed every 
half hour, the total curve was given empirically, and is exhibited in Plate X. fig. 1., 
the curves being thrown into two groups (according to hours of moon’s transit) to 
avoid confusion. At Plymouth, the time and height of high water were observed, and 
also the time of passing two horizontal lines (M and N) situated near mean water 
(at nine feet ten inches and ten feet ten inches respectively, above the zero of the scale). 
By this means five points were given in each curve, (the observations being arranged 
according to the hour of transit), and the curves were drawn by the eye through 
these points, as may be seen in fig. 2. 
At Bristol, the tide-gauge did not allow the rise and fall to be observed more than 
half way down from high water to mean water ; but there is no doubt that in its ge- 
neral law of variation, the curve approached nearly to the figure of sines. 
(2.) But in all these cases, more or less, there is a deviation of the empirical from 
the theoretical form, which deserves notice. Instead of being a symmetrical curve, 
MDCCCXL. 2 L 
