8 T. H. Havelock. 
two parts of the disturbance in each case. We require also the following 
results 
7/2 Kod 
| sip (coq sec 0) d0 = — = J Yo( d= Po (xan) (25) 
0 0 
where the P functions, which have been used previously in wave analysis, are 
defined by 
ar | 2. 
Fon (P) = (— 2)" | cos®” 9 sin (p sec 0) dO 
0 
tr | 2 
Ponsa (p) =(— 1)" | cos*"*1 § cos (p sec 6) dO. (26) 
0 
We have also 
{" a0 I 1 — cos (ky mq sec 8) shin 
0 0 m (1 + m) 
= a dt {" Tq (mt) dm 
2 0 0 1 + m 
ao} 
=F [Go —Yo hd =F Qu), (27) 
using the notation introduced by Wigley for this part of the disturbance. 
Retaining q for points in front and q’ for points behind the origin of a dis- 
turbance, so that g’ = — q and q, q’ are both positive, we find after collectmg 
these results that 
FQ) =—s— Qo (ko), 9 > 0 
0 
acho Tr Qo (Kog’) + = Po (ky), q > 9. (28) 
The complete surface elevation may now be found from (23), (24) and (28). 
The Q terms represent a local disturbance which is symmetrical fore and aft 
for the system as a whole, while the P terms give the wave disturbance in the 
rear of each of the points --a, +l. 
If M is put equal to wb/2x, these results will be found to agree with those 
for the model examined by Wigley in the paper quoted above, and reference 
may be made to it for a detailed comparison with experimental results. 
It should be noted that the method used in (23) and (24) for separating the 
disturbance into four parts is reflected in the artificial character of the local 
disturbance associated by (28) with an isolated point g=0; the function 
Qp is zero at its origi and increases indefinitely with distance from it. The 
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