531 T. H. Havelock 
boro = —q,>+ 0% — U9, — rely Wind mad 
bis = — 442 
bis = $9192 
biz = 49293 — 591°G2 
big = — 490° + 19342 — 49,9093 + 249394 
bi = —4 Gog? + £91 9293 — $91°G2 + 21192 — 37919394 + 2854445 
ba = — €I3 
bes = $4193 
bes = — 441°43 + 729244 
bss = — ha 
bsr = rtd 
Dug = — T2095 (24) 
4—Consider now the forces acting on the cylinder per unit length. The 
pressure is given by 
p/e = const — gy — 44°. (25) 
The term in gy gives the usual buoyancy, equal to the weight of displaced 
liquid, as part of the vertical force on the cylinder. Apart from this 
term, let X, Y be the resultant horizontal and vertical forces on the 
cylinder in the positive directions of Ox, Oy. Then, by the Blasius 
formula, we have 
LA ati (ayy a, (26) 
taken round the circle | z| = a. 
We note that —X will be the force known as the wave resistance, 
while Y is the addition to the upward force of buoyancy arising from the 
fluid motion. The value of the integral in (26) is 27i times the residue 
of the integrand; with w given in the form (15), and, using (16), this 
gives 
I ee 1 
X — iY = 2npz ae 1D), DP va (27) 
Using (14), we have the result 
X— iY = — InpCai {11.25 ,b% + 2.80,0% + ... 
ap BW ap Wb bese oe @8) 
This may be expanded in powers of y, that is of «a, by substituting from 
(23) and (24), the results given there being sufficient to include the term 
425 
