418 T. H. Havelock 
Taking mean values, this gives 
P = toph* sin? a. (32) 
7. We proceed similarly for any fixed cylinder of infinite draft with 
vertical sides; it is not necessary to examine the second order terms for the 
surface elevation, and we assume that the velocity potential is correct up 
to that order, or at least that any second order term is purely periodic. 
Consider the solution for the circular cylinder which was given in § 4. 
We write it as 
@ = (gh/c) e(Lcosot— M sin ot), | 2) 
¢€ = —h(Lsinot + M cosot), | 
where L, M are functions of r, 6 which may be obtained from (17). 
At any point on the cylinder we have 
p = F(t)—gpz—gphe*(Lsin ot + M cos at) 
— (p/2a?) e={x?a?(L cos ot — M sin ot)? + (L’ cosot— M' sin ot), (34) 
with r = a, and the accent denoting 0/00. 
We integrate with respect to z from — oo to € and expand to second order 
terms; then for the resultant force we multiply by acos6d6 and integrate 
round the circle. 
It is readily seen that the first order term in the additional force is a 
periodic effect of amount 
4gph J;(ka) sin ot + Y‘ cos ct 
K? J1?(Ka) + Y{?(ka) 
(35) 
From the quadratic terms we get, after taking mean values, the steady 
additional force 
ip tapha| fave (L24M")} cos 0dé, (36) 
0 
Oy ee) 
where Ib RINE (+, PAO. cos 8), (37) 
77K \ 1 
and b, = i”/H,®’ (xa). 
We have 
Hy C) 4 ok 
[iz oF M?) cos 0d@ = Ae = (Ono ay DSc) 
(38) 
P) 12, Be! Sa * 4 pe 
[az +M ) cos 0d0 ap a n(n + 1) (6, bF 1+ Op, Pnsa)> 
479 
