585 T. H. Havelock. 
A similar integral which we require in (4) can be derived by differentiation, 
and we have 
/2 9 1 7 
3 —Bsec?> =F Ze# {1-3 See De ale 4 
; BOS Gee $ r P Tea a8 at 
683 Dai -@ 3 4 Dae = ee } 10 
* 30720" 7 Ce tage UE ae 2 ESB ee OY) 
For large values of 8, asymptotic expansions can be found in the usual 
manner by transforming the integrals. They are 
alg 1, 189 1 3465 1 
5 fe-Bsec?> dd, ~~ 728-3 (1! eS at I aN cee Net ees Se 
[; cos° de ¢ 578 e Age 32 BF 128 B 
315315 1 
ee 1 
2048 Bt ) 
“fe ihe 51-1031 4725 1 
3 he-Bsec?? dd — — 773 B73 “(12 Se ed Sg 
i. coro Piriataen? 16 92 6? 384 63 
4 3638251 _ : (12) 
6144 B# 
On applying the expression (4) to numerical cases of interest, it was found 
that the integrals we have just considered were required over a range inter- 
mediate between those suitable for the series given in (9)-(12). It was, 
therefore, necessary to calculate the values of integrals such as (5) directly 
by numerical methods. It was sufficiently accurate to evaluate the integrand, 
for each value of 8, at intervals of five degrees throughout the range of 
integration, and then to use Simpson’s rule. The series given above were 
used for checking and supplementing the values so obtained. 
For the purposes of expression (4) the results were collected in tables and 
graphs of the integrals 
1/2 
I= | (1 — e= 28°74)? cos? Jd, (13) 
0 
L= [a eos! Preost pide. (14) 
The graphs are shown, on « as base, in fig. 1. 
It can be seen from (4) how the mean resistance, apart from superposed 
interference effects, depends upon the integrals I, and I;; and since « is 
gd/c?, the curves in fig. 1 show how the effect of finite draught becomes 
appreciable when the wave-length is comparable with the draught, the ordinates 
falling off rapidly in value after that as « becomes smaller. 
233 
