521 Prof. T. H. Havelock. Some Cases of 
Wi, m(a); it belongs to the logarithmic case for which a general expansion is 
not available. 
In general form the graph of the resistance is very similar to that of the 
circular cylinder. 
Circular Cylinder. 
2. The steady state for uniform motion of the cylinder may be attacked 
directly, as in Prof. Lamb’s solution, but we shall adopt his suggestion of 
building it up from simple oscillations. Take the axis of x in the free 
surface of the water, and the axis of y vertically upwards. A circular 
cylinder, of radius a, is making small oscillations parallel to Ox with velocity 
ccos at, the axis of the cylinder being horizontal and perpendicular to Oz, 
and the mean position of the centre being the point (0, —f). A first 
appproximation when the depth / is sufficiently large is found by ignoring 
the surface effect altogether and putting 
p= caa(afr2yext; rPr=e+ytfy. (1) 
This satisfies the boundary condition at the surface of the cylinder. For the 
next step, add a term X, to the velocity potential so as to satisfy the 
conditions at the free surface, but ignoring meantime the disturbance 
produced thereby at the surface of the cylinder. The term X; must) be a 
potential function and it must satisfy the condition for deep water, namely, 
0X, /dy = 0 for y = —co ; these conditions are fulfilled by 
2G = eet a(x)ecsin xx dk, (2) 
0 
where « is a function of « to be determined. This form is chosen because we 
can satisfy the conditions at the free surface by using an equivalent form 
for (1), since 
x fr? = | “erotNsinecde;  ytf>0. (3) 
0 
The surface elevation is expressed similarly by 
tee \,2 (ic) sin «x de, (4) 
0 
In order to keep the various integrals convergent, we assume that the 
liquid has a slight amount of friction proportional to velocity; in the sequel 
the results are simplified by making the frictional coefficient ~ tend to zero. 
In these circumstances the pressure equation is 
) 
p/p = const. + Se —9y+ub—F dae (5) 
Hence the conditions at the free surface are, neglecting the square of the 
velocity, 
d¢/dt—gy+ ubd=const.; —dd/dy=dn/dt. 
