392 T. H. Havelock. 
about ¢ = 4/(gf). On the other hand Y is relatively large at lower velocities, 
and changes sign at about c = 0-84 4/(gf). The wave resistance arises from 
the flow of energy in the regular waves to the rear of the cylinder, while the 
vertical force is associated more with the surface elevation near the cylinder. 
The surface elevation immediately above the centre of the cylinder is given by* 
n = (2a*/f) {1 — ko feos li (e%S) }, (22) 
and for comparison this is shown on the figure with an arbitrary scale for the 
ordinates. Doubtless the variation in the vertical force with the velocity 
is connected with the mean curvature of the lines of flow in the neighbourhood 
of the cylinder. It may be noted that the usual approximation for the pressure 
condition at the free surface involves neglecting the square of the slope of the 
surface ; this would not affect the present approximation but would come into 
the next stage involving higher powers of the ratio a/f. 
4. Obviously in a complete solution the fluid pressures on the cylinder 
cannot give rise to any couple. As the method of successive images amounts 
to an expansion of the solution in ascending powers of a/f, it is worth while 
verifying that the couple is zero at each stage of the approximation. With the 
images specified up to the present the couple comes from the interaction of the 
doublet at C with the line distribution to the rear of C,. Using the result (12), 
this gives a moment 1 
Kop — ae = 20) 
4akopc?a* 
dp, (23) 
sin ( 
i 
0 
p? + 4f? 
which, on substituting for 0, gives 
tnnapcta| Apf sin Kop — (p2 — 4f2) cos Kop oh (24) 
( ° 
0 (p? i; 47°)? 
This can be evaluated, and its value is not zero. But it can be seen that we 
shall get a contribution of the same order, in the radius a, from the next stage 
of successive images. The next set of images is internal to the cylinder and 
consists of a horizontal doublet of moment —ca*/4f? at the inverse point C, 
whose co-ordinates are (0, —f-+ a?/2f), together with a continuous distribu- 
tion of doublets on a semi-circle described on CC, as diameter. At a point 
on this semi-circle whose co-ordinates are 
tg mt fie 
joe ae Ay [Dar Ayr 
*Roy. Soc. Proc.,’ A, vol. 121, p. 517 (1928). 
302 
