THEORY OF SHIP WAVES AND WAVE RESISTANCE. 17 
This distribution of doublets gives over the plane y = 0 a normal distribution of 
velocity of amount 27/82. Taking the plane y=0 as the median fore and aft 
plane of the ship, and taking the ship's surface to be given by y= F(a, z), we have, 
with the assumptions in the text, to substitute 27dy/5x = cd5F/5x in (10) to obtain the 
wave resistance. The curves of Fig. 4 for the form of the water-plane section are 
particular cases of the equation 
b x2 \ ( ( a?) ) 
= an (t= =) 1 1-422 n= =) es ihsy. (LL) 
Here 2/ is the constant length of the ship and 4®J the constant area of the 
water-plane section ; the beam is 2b(1 — 1d2)/(1 — 4d?) The four models are the cases 
d= 0, 1, 1-25 and 1-5 respectively. Evaluating as far as possible the integrals in (10) 
for the form given in (11) we obtain 
512gpb% diate et 
(1 — 4d2)p =[ x a(l "+ 48(1 + 242 - 4d rs 
da? ds 2 
+ Se — + 19984 — 4 4(1 — 3a2)"P, — =(1 — 4424 4d4)P 
p p ( A) a 3 gas)P, 
4 32 64 
+ sell — 6d" + $dtjP, — pail — 30)P, peel 249, 
‘ 4 4 
Ps + Se, | SES Mee rede yi a Mee eee ak (TD) 
where p = 2g//c? and the functions P are defined by 
7 
P., (p) = (= uf ‘ cos?” @ sin (p sec d)dp 
0 
T 
12 ana) = (oper i ; cos"”*! ¢ cos (p sec )dp. 
0 
After preliminary computation of these new functions, it was possible to calculate 
R from (12) for the four given values of d and for sufficient values of p in each case to 
give the curves of Fig. 5. (Proc. Roy. Soc. A., 103, p- 571. 1923.) 
6.—The equation of AB in Fig. 6 is 
y=b{ 1 —(e — mae } Be Ry atRe wiiate tn. 1125513) 
In this case the integrals of (10) give, with the same notation, 
512 gpbl Z 4 
R =a [33 BiG Tar = P, (2p) + 2 Fs (bp) 
2 2 
+$P, (p,) - 3 P,(p,) + Poe (p,) + Pp P, (p,) 
4 2 
. ae (p2) + pits (Ps) |. . . . : . : (14) 
where p = 29l/c*, P1=9(2k +21)/c2, pog=9(2k+1)/c2, pg=2gk/c?2. 
The curves of Fig. 7 were obtained from this formula, with 7=80, for the cases 
g/c?=0'1, 0:0625, 0:05, 0-045, 0:04125 respectively. (Proc. Roy. Soc. A., 108, p. 77. 
1925). 
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