279 T. H. Havelock. 
The amplitude C has a maximum at the speed 4/(2ga) ; and it appears from 
the table that the second approximation: increases the amplitude below this 
velocity and diminishes it at higher velocities. It seems that thé rearward 
displacement, given by &, also has a maximum, amounting to about two-thirds 
_of the radius of the cylinder. 
7. It is clear, from the form of the expressions for the surface elevation, that 
the accuracy of the first approximation increases rapidly as the depth of the 
cylinder is increased or as we take relatively smaller velocities. Without 
pursuing the calculations in this direction, we shall take one other case which is 
not quite so extreme as in the previous section. We take the depth of the 
centre to be three times the radius; the data are now 
J = 3a; k= 2of; h = xat/4f = k/72. (33) 
In this case, instead of (30), we have 
nla = 2ke—* sin xox — Aprke-™ sin xox 
+ gynke—™ (A cos xoz — B sin oz). (34) 
The following table shows the values of A and B, with h = k/72, calculated 
for convenience at the same values of k as before :-— 
k | 10 | 8 6 | 4 2 | 1 | 0-5 
A 0-008 0-033 0-137 0-540 1-747 2-311 1-898 
B —0-324 | —0-428 —0-626 | —0-911 —0-644 0-663 1-732 
With the same notation as in (31) and (32), the results are given in the 
following table :— 
c// (9a). | Cc | D | E/a. 
0:77 0-141 0-149 0-001 
0-87 0-307 - 0-329 0-006 
1-0 0-626 0-677 0-024 
1:22 1-134 1-222 0-088 
1-73 1-541 1-558 0-295 
2-45 1-270 1-214 0-409 
3-46 0-816 0-802 0-324 
276 
