523 Prof. T. H. Havelock. Some Cases of 
writing V for ,/(g/«) whenever it serves to simplify the notation. Hence 
the velocity potential of the subsequent fluid motion after the cylinder has 
been given a small displacement at time 7 is 
0 
h = 2ca*Bre-mtt—n [ nV e-*0-2) sin ex sin eV (t—7) de. (12) 
Finally we obtain the velocity potential for a cylinder in uniform motion 
by substituting «+c(¢—7) for x, noting that hereafter x will refer to a 
moving origin immediately over the centre of the cylinder; we then 
integrate with respect to 7 from the start of the motion up to the instant in 
question. We could in this way obtain results for any stage of the motion, 
but we limit the discussion to the final steady state ; for this we take —oo as 
the lower limit in integrating with respect tor. Before writing down the 
result, we must remember to introduce the integrated effect of the original 
momentary doublet in (1) and its negative image, which were not included in 
(11); these clearly add up to steady doublets. Hence we find for the steady 
state 
4 = DD Beat | e-ett-0 (A sin xe +B cos xz) de, (13) 
0 
where D represents the doublet ca*x/r? at the point (0, —/), D, an equal 
doublet at the point (0, f), and 
oa — EV (V 40) eV (V—e) 
eV +eP tty? (Veh + dp 
Aig =) a DEN Sats oe ae BSN (14) 
eO(V—cp +i? (V+ e+e 
3. Before proceeding further we may obtain the surface elevation from (13) 
for comparison. The surface condition is now 
—0¢/dy = On [Ot — —con/ ox. 
Hence we have 
n = 20?f (a? + f2)—20° | *(A cos xe—B sin wx) e-* de, (15) 
0 
in which x = g/c. Further, since yu is to be small, we may omit irrelevant 
terms and put 
A = ho (1e—Ma)/ (0—(4a+ in /0)} {1e—(0o— ip 0), 
B = wo (u/0)/ {1e—(Ho + iu] e)} {«—(xo— ipo}. (16) 
The integral in (15) can then be written as 
C) enikx eiKe 
—K«f 
I, (an See (ie (17) 
122 
