258 THE REV. W. WHEWELL’S RESEARCHES ON THE TIDES. (SERIES XII.) 
the rise and the fall taking place at the same rate, the empirical curve of rise and 
fall is unsymmetrical, the fall being more rapid than the rise in general, though some- 
times the reverse occurs. By this means the summit of the curve of rise and fall 
is displaced, and is posited to the right or left of the position given by bisecting the 
interval of the times at which the surface, in rising and in falling, passes some lower 
line (as the line of mean water). This displacement of the summit I have taken at 
Liverpool for two lines (M and N) at the heights of fifteen feet and seventeen feet 
above the zero ; at Plymouth for two lines (M and N) which, as above stated, a?*e at 
nine feet ten inches and ten feet ten inches above the zero. In these cases the lines 
M and N are very near mean water. At Bristol, as I have said, the lower observed 
height was half way from mean water to high water. 
The displacement of the summit being arranged according to the hour of moon’s 
transit, there is no obvious agreement in the form of the result for different places. 
Fig. 3. and 4. give the curves which express these results for Plymouth and Liverpool ; 
Bristol is not readily comparable with these, from the nature of the observations. 
(3.) The variations in the length of the tidal half-day (from low water to low water) 
were also examined. It appeared that at Liverpool and at Plymouth the variations 
in this element followed nearly the law indicated by the theory ; which, indeed, may 
be considered as having been previously sufficiently proved by the near accordance 
of the observed with the theoretical times of high and low water ; an accordance 
which had already been established by the discussion of the observations at Liverpool, 
London, and other places. 
(4.) The same may be said of the variations in the height ot high and low water 
during a semilunation. It had already been repeatedly shown, that (if we assume a 
proper value of c) these, as observed, agree with great accuracy with the theory, at 
every place yet examined. 
(5.) Plence, as I have said, the intersections of the curves of rise and fall belonging 
to different hours follow, in the curves given by observations, nearly the theoretical 
law. Thus these intersections all take place nearly at mean water; but occur a 
little higher and nearer high water than the height and time of mean water. 
(6.) But yet the want of exact symmetry in the empirical curves occasions some 
peculiar features in the intersections of these curves. Thus, at least for many of the 
hours of transit, the intersections of the curves of rise and fall are higher and nearer 
the time of high water in descending than in ascending. This may be seen in the 
Liverpool curves, in which many of the ascending intersections fall very near the line 
M, while the corresponding descending intersections fall near the line N. 
(7.) Since the time of ascending is greater, and the height ascended less, at neaps 
than at springs, it is plain that the velocity of ascent must be greater, and the time 
of ascending a given space less, at springs than at neaps. The same is the case of 
descent. And hence the times of ascending from M to N vary, and this variation 
appears eminently in the rate of the surfaces ascending or descending about mean 
