527 T. H. Havelock 
tance; the corresponding extension of the solutions would require more 
complicated analysis than for the two-dimensional case, but it seems 
probable that the general deductions on the range of applicability of the 
approximate formulae would be of a similar character. 
2—Consider the two-dimensional fluid motion due to a fixed circular 
cylinder, of radius a, placed in a uniform stream of great depth, the axis 
of the cylinder being at a depth f below the undisturbed surface of the 
stream. Take the origin at the centre of the circular section, with Ox 
horizontal and Oy vertically upwards, and suppose the stream to be of 
velocity c in the negative direction of Ox. We write the velocity potential 
of the motion as 
$= CX + $y. (1) 
To obtain a solution which gives regular waves to the rear of the cylinder, 
we adopt the hypothesis of a frictional force proportional to the deviation 
of the fluid velocity from the uniform flow c, thus introducing a coefficient 
1" which is made zero after the various analytical calculations have been 
effected. The pressure is then given byt} 
E = const — gy + why — 448. (2) 
If 4 is the surface elevation and we make the usual approximation for 
small surface disturbances, we have 
= ee Oe, nt 
ees Pagtnrs taeda (3) 
Hence, from (2), the condition to be satisfied at the free surface is 
Fly pg, es fy Oe op. ye (4) 
where ky = g/c*, and p = p'/c. 
We may regard ¢5 as made up of two parts, one part having singu- 
larities within the circle r = a, and the other having singularities in the 
region of the plane for which y > f. The first part is the potential of a 
system of sources and sinks, of total strength zero, within the circle, 
and can clearly be expressed by the real part of a series 
DAZ", (5) 
T Lamb, “Hydrodynamics,” 6th ed., p. 399. 
421 
