[Reprinted from the PRocEEDINGS oF THE Royat Soctury, A, Vol. 132.] 
The Wave Resistance of an Ellipsoid. 
By T. H. Havetocr, F.R.8. 
(Received May 9, 1931.) 
1. Inarecent paper* it was shown how to obtain, to the usual approximation, 
the wave resistance of a solid of ellipsoidal form submerged at a constant depth 
below the surface of water and moving horizontally with any orientation of 
the axes ; and explicit calculations were made for prolate and oblate spheroids 
moving end-on and broadside-on. The present note is a brief study of an 
ellipsoid with unequal axes, moving in the direction of the longest axis. It had 
been intended to examine numerically in some detail the effect of different 
ratios of the axes upon the resistance-velocity curve; but the necessary 
computations would have been lengthy, and the main results of interest may 
be seen from the form of the expressions obtained for the wave resistance. 
In the discussion attention is directed specially to cases in which the ratios 
of the axes are similar to the corresponding ratios for a ship. 
2. It is convenient first to evaluate some integrals which occur in the 
analysis. 
Consider the integral 
ao ye\i2 
A= [{(i- 7 -*) cos ax cos By dx dy, (1) 
m= n? 
taken over the ellipse 
Gil UES 2 
#+4- (2) 
Putting z= msin ¢ cos 0, y = n cos ¢, we obtain 
A =mn [ if sin? ¢ sin? 0 cos (ma sin ¢ cos 0) cos (n8 cos 6) d0d¢. — (3) 
0/0 
Integrating first with respect to 0, this gives 
A = (xn/a) Ik cos (nB cos ) J, (ma sin ) sin? ddd 
0 
a 
= =. i J, (ma sin $) J_4/2 (nB cos ¢) sin? fd cos’? ddd. (4) 
0 
* «Proc. Roy. Soc,,’ A, vol. 131, p. 275 (1931). 
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