502 Prof. T,. H. Havelock. 
As in similar plane wave problems, this integral can be modified by inte- 
grating round a suitable contour in the plane of a complex variable; the 
expressions then divide into two types according as the integrand has or has 
not a pole within the contour. The surface disturbance corresponding to (10) 
is seen then to consist, in general, of a surface elevation symmetrical with 
respect to tne line xcos¢+ysin d = 0, together with a regular train of waves 
in the rear of this line; but the latter part only occurs if c?cos?’@<gh. In 
evaluating the wave resistance by (5) for the symmetrical distribution (9), 
we see that we need only consider the regular train of waves. By calculating 
the residue of the integrand in (10), collecting the results and finally making 
» zero, we find that the regular waves, when they occur, are given by 
___ Am Aciste sin (xa!) (11) 
g sec? h (c? —gh sec? o) + x2cth’ 
where « is the root of 
Ke?—g sec’ tanh ch = 0; gh sec? p > c?. (12) 
From (5) and (11), the contribution of this element to the wave resistance is 
An Ac?ke—* cos b i — ACOs (ce) dex! 
g sec? dh (2 —gh sec? $) + K*cth oe! | oa? +y2+ PP 
223 p—2xl 
te emai Ke **' cos h (13) 
g sec? b (ce? —gh sec? b) + x*cth 
Summing for the different elements, from (13) and (7), we have finally for 
the wave resistance 
_ 4 Ate? pr? Ke *! sec hdd 
R= p i g sec? h (? —gh sec? h) + x2cth’ se) 
where « satisfies xc? = g sec? ¢ tanh ch, and the lower limit ¢o is given by 
goo = 0, for &<gh; go = are cos (gh/c?)V?, for c? > gh. (15) 
4. We may notice, in the first place, that (14) reduces to the expression given 
previously for deep water; making >, we find 
ar/2 
1 = (A4rrg?A?/pc®) | sect  e-? (lle?) see 4 de 
0 
A293 
= EES {i Gy — Em}, 6) 
where « = gl/c?, and the result is expressed in terms of Bessel functions of 
which Tables are available. For finite values of the ratio 4//, the value of R 
for given values of ¢ can only be obtained from (14) by numerical and 
graphical methods. After some preliminary trial, the following plan was 
195 
