Theory of Wave Resistance. 343 
This result may also be put in the form 
1 = Seeks (e (P2 + Q2) sec? 6 d0, (16) 
with i 
al = | o er see 8 C8 F(a cos 8 + y sin 0) sec? 6} dS. (17) 
sin 
In (17), the co-ordinates (h, k, —f) have been replaced by current co- 
ordinates (x, y, 2); since the sources are within the liquid, z is negative over 
the surface S. 
Doublet Distribution. 
4, A surface distribution of normal doublets could be obtained by general- 
ising an expression for any two doublets, but it can be deduced directly from 
(16) and (17). We have simply to regard the surface S in (17) as a double 
sheet with source densities c and —o respectively, and then proceed to the 
limit in the usual manner. The required result is obtained by applying the 
operator 
a) 0 a) 
[es = = 
Be ay ae! 
to the expressions in (17), (J, m, n) being the direction of the normal to the 
surface. If Mis the doublet moment per unit area, the axes being everywhere 
normal to the surface 8, we obtain, in this way, the wave resistance 
B= Suelo | (P2 + Q2) sec? 6 d0, (18) 
—in 
in which 
P= | M ese? {_ (J cos 6 + msin 8) sin (ky @ sec? 8) 
+ n cos (ky @ sec? 0)} dS 
Q= | M ese? {( cos @ + msin 8) cos (ky@ sec? 0) 
+ n sin (k gw sec? 6)} dS, (19) 
with o = xcos 0 + ysin 0. 
These expressions may be put into various alternative forms, and, of course, 
may be simplified when the surface distribution is symmetrical with respect 
to the co-ordinate planes. It may be remarked that an expression given 
previously for the wave resistance of any two finite doublets in given positions 
may be deduced as a particular case of these results. 
