238 T. H. Havelock. 
terms factored by sin (p sec ¢) or cos (p sec ¢). For large values of p, we 
have an asymptotic expansion of any of these terms in the form 
/2 5 
[; cos" gf . eiPseod— BPE Hd g~ pF (ay + apt + agp? +...). (17) 
0 
Moreover, in practice, the interference effects concerned are prominent for 
larger values of p, say, for the range 10 to 40. Now from (10) we see that 
the expansion of the oscillatory terms would begin with a term of order p-}, 
while from (16) the lowest term is of order p-?. It follows, therefore, that the 
interference effects have been largely eliminated by the alteration made in the 
form of the model. It may be noted that the alteration is rather extreme if 
considered as an illustration of practical conditions, in that the after end of 
the model is cut away completely to zero angle ; this accounts for the complete 
absence of the term in p-* in the expression for the second case. 
6. To examine the matter graphically, it is easier to consider a model of 
infinite draught, and of small ratio of beam to length, in the manner used in 
previous papers. The model is assumed to be symmetrical about a longi- 
tudinal vertical plane. Take Oz horizontally in this plane, and let Oy be also 
horizontal. The form of the horizontal cross-section of the model is constant ; 
if its equation is 
y = F (2), (18) 
for positive values of y, the approximation consists in taking the doublet 
distribution of (4) so that 
Qn Op/ex = coF dz. (19) 
Integrating (4) by parts with respect to h and h’, substituting from (19), and 
also integrating with respect to f and f’, we have 
= “eee dh (¢ dh’ i cos jee) sec 6) cos ¢ dd. (20) 
We wish to contrast two models which have the front half the same, but with 
the rear end smoothed off to a finer point in one case than in the other. We 
shall take the section of unsymmetrical form to be given by 
y = (b/48) (lL—2) (2140); —W<a<l. (21) 
For the symmetrical model we shall take the front portion to be given by (21) 
for x positive and by the corresponding expression for xz negative. The model 
in one case is of length 21 and in the other of length 3/. The cross-sections by 
a horizontal plane are shown in fig. 1. 
245 
