Wave Resistance 464 
within a distinct closed surface. For convenience, we shall limit the 
discussion to a distribution in the vertical plane y = 0. We suppose that 
the total distribution o of the previous section can be divided into two 
distributions 6, o,, each representing a solid body and one being aft 
of the other. The resistance for either body is given by the same general 
expression (3), the integration being taken over the corresponding partial 
distribution. For instance, for the resistance R, of the body o,, the 
velocity u at any element of c, will be that due to the rest of o, and to os, 
and the integration is to be taken over o,. The velocity potential is given 
by (2) with o = o, + oy. It is convenient to regard (2) as made up from 
the following parts: the uniform stream c, the given distributions o, and 
Oy, distributions —o, and —o, over image positions above the free surface, 
and finally a part represented by a certain integral taken over the dis- 
tribution o, + oz. 
Consider the contributions of these parts to the value of R,. The 
uniform stream gives no resultant effect as we suppose the integrated 
source strength of o, to be zero. We have now a resultant force from the 
mutual actions between o, and o2, given numerically by ; 
h, — hy 
4ro \| o, dh, df, \| NE ee Serer | (il) 
the sign depending upon whether o, is in advance of oc, or to the rear of 
it. It may be noted in passing that this corresponds to the apparent 
repulsion between two bodies, one behind the other, in a uniform stream 
of infinite extent. There is also a similar resultant due to the actions 
between —o, and oj, given numerically by 
Arco \| o, dh, df, \| oy dh, df, levies (12) 
* {Gy = ha)? + (fs + fa)? 
Finally we have the part due to the last term in (2) for the velocity poten- 
tial, and this will be given in the notation of (8) by 
Bicop | | o', dh’, df’, | (o, dh, df, + 6» dhs df») RE sec 0.0, (13) 
J 0 
where F is given by (6) and (7). 
The terms in F represented by the integrals in m will give a resultant 
effect different from zero when summed over the partial distribution o,, 
arising from the part due to o, when summed over o,. From the term in 
(7) representing the regular waves, the part due to o, when summed over 
oc, will give the wave resistance of o, as if existing alone; the part due to 
412 
