492 Dr. T. H. Havelock. 
If A is a constant and independent of the speed v, the graph of R as a 
function of v rises to a maximum and then falls slowly but continually to 
zero as V increases indefinitely. Thus, for an assigned pressure disturbance 
of this type whose magnitude is independent of the speed, there is a certain 
speed beyond which the resistance R continually decreases. 
On the other hand, if the pressure disturbance is that produced by tue 
motion of a floating, or submerged, body, it is clear that it will depend upon 
the speed. Since we may suppose the pressures in question to be the excess 
or defect of pressure due to the speed, it seems a plausible first approxima- 
tion to assume that the distribution is not altered appreciably in type and 
that the magnitude is proportional to v2. Thus if in (11) we make A pro- 
portional to v? we obtain 
R = const. x e~ 29a”, (12) 
The value of R now tends to a finite limiting value as v increases 
indefinitely. 
If the quantity A, specifying the magnitude of the pressure disturbance, 
varies as v", then the graph of R rises to a maximum for some finite value 
of v, provided x is positive and less than 2; the nearer x is to 2 the higher 
is the speed at which the maximum occurs. For the present we assume 
that n is equal to 2; in any case it does not affect the results of a qualitative 
comparison of different types of distribution. 
The scope of the assumption may be illustrated by a certain case. 
Prof. Lamb* has worked out directly the wave-making resistance R due to 
a circular cylinder of small radius a, submerged with its centre at a constant 
depth f, and moving with uniform velocity v; he finds that R varies with 
the speed according to the law v~4e-2a/, Tf we attempt to represent the 
disturbance approximately by some equivalent surface pressure distribution 
the type which suggests itself naturally is 
p=A(P—x)[(7? +2). 
It can be shownf that this distribution, together with the assumption that 
A is proportional to v*, leads to the same law of variation of resistance with 
speed. 
4. In a certain sense the generalisation from a line disturbance to any 
diffused distribution of pressure may be regarded analytically as a case of 
interference ; the final result is due to the mutual interference of the line 
elements into which we may analyse the given distribution. However, the 
idea of interference in ship waves has usually been associated, after the work 
2 
* H. Lamb, ‘Ann. di Matematica,’ vol. 21, Ser. 3, p. 237. 
t ‘Roy. Soc. Proc.,’ A, vol. 82, p. 300. 
or 
