Wave Resistance. 30 
x — hinstead of x, together with a similar expression in M’ and « — h’. Taking 
the real part we have 
es) 
ee cal se a0| cos (ky sin 0) 
16 daa » (Kk — Ky sec? 0)? + pu? sec? 0 
Pe dk, 
P = wsec 0[Me~“! cos {« (x — h) cos 0} + M’e~*!" cos {k (x — h’) cos 0}] 
— (k—ky sec? 0) [Me~“! sin {x (z—h) cos 0}+M’e—"" sin {x (c—h’) cos }] 
(24) 
There is a similar expression for the surface value of 0¢/0z. We now write both 
these in the form (18), omitting terms which from symmetry give zero when 
integrated with respect to 0. We find again that we have only to form the 
quantity F,G, + F,G, and that this simplifies considerably ; the wave resist- 
ance, after this reduction, is given by 
nea I6exya | do) “ee {« (h—h’) cos 6} dc 
0 0 0 (k—Ky sec? 0)?-+ uy? sec? 0 
mf 9 2 9 ’ 2, 
—_ 169% | [M2 e7 2Kof sec t) +M’” e7 2kof sec? @ 
0 
+ Q2MM’e~"(F47) sec* ® cos fi, (h — h’) sec O}] sec Od0. (25) 
The first two terms give the resistance due to the two doublets separately, 
while the third term represents the interference effects. This expression was 
obtained formerly from the analogy between the waves produced by a sub- 
merged sphere and those due to a certain surface distribution of pressure ; it 
was then generalised for any distribution of horizontal doublets in the vertical 
plane y=0.* The method given here can also obviously be extended by 
integration for any such continuous distribution, and confirms the general 
expression used in previous calculations; if M (h, f) is the moment per unit 
area at the point (h, 0, —f), then (25) generalises to give 
R= Were? | df | ar’ | a dh’ | eM (hy f) MW, f”) x 
0 0 J-o Jw 0 
e Ko (F+F') 80" 8 cos {icy (h — h’) sec 8} sec® Od. (26) 
General Distribution. 
5. We can use the same method for doublets with their axes in any directions, 
for we can always obtain the surface values of ¢ and 0¢/0z in the form (20) 
and so can integrate over the surface by means of (24). Begining with a single 
* “Roy. Soc. Proc.,’ A, vol. 95, p. 363 (1919) ; also A, vol. 108, p. 78 (1925). 
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