483 T. H. Havelock. 
For a distribution of this type the expression* for the wave resistance of 
any two doublets generalises into 
RB = 16x! | TH Pee 2naf °° soo5 dO, (17) 
where : 
1B j)™ (x, 2) M(z’,2’) cos {kg (7 — 2’) sec 0} cos {cg (z — 2’) sin 0 sec? 6} 
dx dzdx' dz’. (18) 
From the symmetry of the distribution specified in (15) and (16) we see that 
1P = {J M (a, 2) cos (kg& sec 8) cos (kz sin 0 sec? 0) dx dz, (19) 
where M is given in (16) and the integration extends over the ellipse (15). 
Comparing with (1) and (5), we obtain 
(2n)"2 abeu Isjp [Ky sec 0 (a? — b? + (c? — b2) tan? 0}/?] 
Da _ BIBI 0 BEG ON Osa Cater 25) MAN Urey 
2— oy Ky? sec? O a2 — 62 + (2 — 62) tan? O94 
0 0 
(20) 
From (17), after putting tan 0 =t, we deduce 
(2 nee oy)? (a2 aan b2)3/2 e2kof R 
327?goa7b7c? 
= [te (eg? (a? — 88) (1+ 8) (1+ o?)PP one gy (1) 
0 (lor GARE 
where a? = (c? — b?)/(a? — 6). 
The cases c <6 and c> b have been worked out separately ; however, on 
comparing (14) and (21), we see that the results could both be included in the 
same formal expression with a suitable interpretation of the integrand when 
a and 1 + «/? are negative. 
5. A numerical examination of these results could be made for different 
ratios of the axes a, b, c; certain points of interest may, however, be seen 
from the form of the expressions, keeping in view the analogy with the wave 
resistance of a ship. We note in the first place that the exponential factor 
exp. (—2« ft?) in the integrand means in practice that the greater part of the 
value of the integrals arises from. small values of the variable t. 
An interesting feature of curves of wave resistance and velocity is the 
occurrence of so-called humps and hollows which, on a simple theory, arise 
from interference between bow and stern wave systems. In (14) and (21) 
* «Proc. Roy. Soc.,’ A, vol. 118, p. 32 (1928). 
326 
