Theory of Wave Resistance. 347 
a so-called interpolation formula which, when put into the notation of the 
present paper, is , 
pee — | "(P24 Q2) sec? 6 d8, (35) 
Pu 
me Jing 
al = {| a Get oe {icy (v cos 8 + ysin 8) sec? 0} da dy. (36) 
In (36) the integrations are taken over the section of the ship by the water 
surface, and the surface of the ship is given by an equation z= F(a, y). It 
may be noted that if dS, and dS, are the projections of an element of the 
surface upon the za-plane and the zy-plane respectively, we have 
(dz/0x) dS, = (@y/@n) dS, 
In the limit y > 0, (86) becomes equivalent to (34) under the conditions for a 
ship of small beam. On the other hand, in the limit z > 0, (36) reduces to the 
expression (31) for a pressure distribution with the assumption p= got. 
Without discussing this argument, it may be remarked that (36) 1s a particular 
case of the expressions in (16) and (17) for a distribution of sources over 
the surface of the ship. In the one extreme case, the narrow ship, we 
take co = (c/2z) dy/0x, the sources forming in the limit a plane distribution. 
For the other extreme, the flat ship, a similar approximation would be 
o = (c/27) 0z/ox. But it is only in these cases, when the source distribution 
approximates to a plane, that the normal fluid velocity can be expressed 
simply in terms of the source density ; these expressions do not hold when 
the distribution is on a curved surface or, in other words, when the finite beam 
of the ship 1s taken into account. 
It has been remarked that formule in use at present are in effect special 
cases of the general expressions (16) and (17), with simple approximations to 
the density of the source distribution. If we think of the distribution, appro- 
priate to motion in an infinite liquid, asa suitable first approximation, it might 
be suggested that this should be used over the curved surface of the ship 
instead of the present simple expressions over the vertical longitudinal plane. 
In one sense this would be an improvement, but it is not likely that it would 
give any better agreement with experimental results ; for the more we depart 
from the simple narrow ship the more necessary it is to take into account the 
effect of the wave motion upon the distribution of fluid velocity round the 
ship. 
Instead of attempting to assign in advance a distribution of sources or 
doublets over the surface of the ship, it might be left to be determined, from 
