418 T. H. Havelock 
the point (4,0, —f). Then the surface elevation ¢ due to this travelling 
distribution is given by 
t= =e | Pema 
Die 
7 
a —Kfhine dk 
OO) ee a(l 
SFE es (1) 
7 
where w = (x —h)cos 9+ ysin 0, and the limiting value is to be 
taken as the positive quantity p tends to zero. 
If the form of the ship is given by y as a function of A and f, the usual 
a proximation is to take F (A, f) as equal to c dy/ch. We modify this 
now by supposing that the effective value of ¢ in this expression for 
F (A, f) diminishes from bow to stern; we introduce what may be called a 
reducing factor f(h), so that we shall use in (1) 
FAN=F¢02. 2) 
We have assumed that the reducing factor is independent of the depth. 
It will, no doubt, depend upon the velocity and form of the model, and 
in particular upon the value of the Reynolds number; but, meantime, 
we shall neglect any such considerations. It may even be that, in 
certain circumstances, the factor should allow for an increase of 
apparent efficiency near the bow of the model. However, it appears 
from such experimental evidence as is available that the wave profile 
near the bow agrees fairly well, for simple models, with calculations made 
without any allowance for frictional effects ; so that the chief effect of the 
latter appears to be a reduction in efficiency over the rear portion of the 
model. In view of these considerations, and also to lighten the numerical 
calculations, very simple expressions have been used in the following 
work. Calculations are made for two cases, and in both we assume the 
reducing factor to be constant and less than unity over the rear portion; 
in one case the factor is taken as constant and equal to unity over the front 
portion, while in the other, to avoid possible discontinuities, it is assumed 
to diminish uniformly from the bow to the value which it has for the rear 
portion. 
We shall consider only models of great draught and of uniform horizontal 
section ; for such, (1) and (2) give for the surface elevation 
=o! -P poplar fpr aecs att EE Gichas «tien 
oS 27? | f@ oh di see ae |, Kk — Ky sec? 0 + in sec 0 3) 
3. We consider a model of length 2/ and beam 26, and of symmetrical 
parabolic lines given by 
y=b(1— FPP). (4) 
399 
