519 T. H. Havelock. 
The corresponding velocity potential is 
> Tu 
=e (P, cos A cos B — P, sin A sin B — P, sin A cos B 
0) 
+ P, cos A sin B) e8°°"? cos 0d0. (28) 
With this value of ¢ in (25), we use (8) and (9) to evaluate the integrations 
with respect to y as in §2; and we obtain readily the general result 
RS tae | "(pe + P+ P,2 + P,2) cos? 6 d0. (29) 
0 
The actual calculation of the quantities P for a body of given form is, of 
course, another problem. Methods in use at present amount to replacing the 
body by some approximately equivalent. system of sources and sinks; the 
functions P then appear, in general, in the form of integrals taken over the 
surface of the body. We need not consider these here as the expressions for 
R given above lead to the same results as those obtained previously by different 
methods. 
5. It is of interest to examine a similar problem when the water is of finite 
depth h. It is clear from the derivation of (25) that we may use it in this case 
also, taking the lower limit of integration with respect to z to be —h instead 
of — ©. 
For the simple symmetrical type of free wave pattern given by (4), the corre- 
sponding velocity potential is 
cosh x (z + h) 
Aaa cos (Kx’ cos 8) cos (ky sin 8) cos 8d0, (30) 
p= 2| 7) 
the relation between « and @ being 
kK — Ky sec? 6 tanh ch = 0. (31) 
We shall assume first koh > 1, that is c? << gh, so that (31) as an equation for 
« has one real root for each value of 0 in the range of integration. In evaluating 
(25) we carry out the integrations with respect to y by means of (8) and (9). 
For this we have to change from an integration in @ to one in a variable u 
given by 
u =k sin 6, (32) 
together with (31). The corresponding factor d@/du has now the value 
cos? 6 (coth xh — xh cosech? xh) 
Ko (1 + sin? 6 — kph sech? xh) ~ 
395 
