390 T. H. Havelock. 
image system of this doublet in the free surface of the stream consists of a 
horizontal doublet of moment —ca? at C, the image point (0, f), together with 
a continuous line distribution of doublets along a horizontal line through C, 
in the negative direction of Ox. At a point (—p, f) on this line the moment 
of an elementary doublet is 2«,ca*dp, where xy =g/c?, and the axis of the 
doublet makes an angle x)p — 47 with the positive direction of Ox.* We may 
stop at this stage meantime. 
In the notation of the previous section the external system (s) consists of 
the uniform stream and the image system just specified ; the internal system 
(r) is the doublet ca* at the point (0, —f). 
For the wave resistance R, we have from (9) 
a ial * cos (kop — $x — 36) 
R = —X = 8rpx,c?a* if (2 af dp, (13) 
sin 0 = 2f/(p? + 4f?)?; - cos 0 = — p/(p? + 4f?)?. 
where 
This gives 
R= Sica | ee de, ag 
0 (sr Ee 
“p sin Kop + 2f cos 
[ee 
0 
D4 af? (15) 
by differentiating twice with respect to f; and we obtain 
R = 4r2gon,2ate 2, (16) 
in agreement with other methods. 
Turning to the vertical force, if we calculate it from the expressions in (9) 
we shall obtain the hydrodynamic part depending upon the velocity. There 
is also the hydrostatic part gpxa*, arising from the term gpy in the expression 
for the pressure ; and in addition there is the weight of the cylinder itself. 
We may assume the cylinder to be of the same density as the liquid, and then 
the calculation will give the total vertical force. 
Measuring Y vertically upwards, the contribution of the two finite doublets 
at C and C, is, from (9), 
—mpc?a4/2f3, (17) 
*‘Roy. Soc. Proc.,’ A, vol. 115, p. 271 (1927). 
300 
