FALL OF THE TIDE IN THE RIVER THAMES. 
5 
On these expressions we may make the following remarks : — 
1st. The mean height of water (understanding by the mean height that part of the 
expression for the height which is independent of sines and cosines of periodical 
terms) is, at Deptford, not the same for spring tides and for neap tides. The mean 
height in the average of the high tides is 13 ft. 3 in. below the top of the wharf-wall, 
and in the average of the low tides is 13 ft. 10 in. below the same point ; or the mean 
height in high tides is greater than the mean height in low tides by seven inches. 
The corresponding difference in the whole range of the tide is about four feet. 
2nd. The curves representing the law of rise and fall of the water are different for 
high tides and for low tides, and both are sensibly different from the line of sines. This 
is evident from the algebraical expression, which contains other terms than those de- 
pending on the sine and cosine of the simple phase : but it will be more evident to the 
eye on the comparison of the curves as graphically traced. In Plate I. the strong 
line represents the law of depression of the surface of the water through every instant 
of a tide, the horizontal abscissa representing the time or rather the phase, and the 
vertical ordinate measured downwards representing the depression (as taken from the 
Table, page 4) for division A (neap tides) : the faint line is a line of sines, whose 
highest point coincides with the highest point of the tide-curve (supposed to have the 
converted depression O013 for phase 350°), and whose lowest point is depressed as 
far as the lowest point of the tide-curve (supposed to have the converted depression 
1‘982). In Plate II. the curves are similarly traced for division B ; the highest point 
of the tide-curve is supposed to have the converted depression 0'008 for phase 355°, 
and the lowest point is supposed to have the converted depression T982. 
3rd. If we investigate the motion of a very long wave (as a tide-wave) in a rectan- 
gular canal whose section is everywhere the same, on the supposition that the extent 
of vertical oscillation bears a sensible proportion to the mean depth of the water : 
putting k for the mean depth, v 2 = g k, and X for the horizontal displacement of 
any particle (x being its original horizontal ordinate), we find the following partial 
differential equation : — 
This equation cannot (it appears) be solved in finite terms, but it may be solved ap- 
proximately by successive substitution. Putting it in the shape 
d 2 X 
dt 2 
— V 2 
d 2 X 
dx 2 
= V * 
d 2 X 
dx 2 
f „ dX , „ / d X\ 2 I 
and first neglecting the second side of the equation entirely, and solving without it ; 
then substituting the solution (adopting that form of function which is adapted to 
the sea-tide at the mouth of the canal) in the first term on the second side, and sol- 
ving again ; then substituting the solution in the two first terms on the second side 
and solving again, &c., we find as many terms as we please for X. Then the vertical 
