278 T. H. Havelock. 
of motion. For a prolate spheroid the system is a line distribution along Ox 
between x = + ae, with axes parallel to Oy, and of moment per unit length 
A’u (a?e? — 2°), (12) 
where 
A’~1 = 2¢ (2c — 1)/(1 — e2) + log {(1 + e)/(1 — e)}. (13) 
For an oblate spheroid the system consists of doublets parallel to Oy, over the 
circle 
pa0; PHP aber, au) 
and of moment per unit area 
Blu (b7e — y? — 2*), (15) 
where 
Bt = nf{e (1 + ¢)/(1 — e)2 — sin“ e’}. (16) 
For e = 0, all these distributions reduce to the finite doublet at the origin 
appropriate to the motion of a sphere. 
4. Consider now the wave resistance when an ellipsoid is wholly immersed 
at some depth in water and is moving with constant horizontal velocity ; we 
obtain the first approximation for the resistance by replacing the ellipsoid 
by the image system which was discussed in the preceding section. The 
resistance is derived from the doublet system by expressions which have been 
given previously ; in particular, reference may be made to an expression for 
the wave resistance corresponding to two doublets at any points in the water 
with their axes in any given directions.* We shall not quote the general 
result, as we require here only the case in which the doublets have their axes 
parallel to the direction of motion. Take the origin O in the free surface of 
the water, Oz vertically upwards ; for a doublet of moment M at the point 
(h, k, —f) and a doublet M’ at (h’, k’, —f’), both axes being parallel to Oz, the 
direction of motion, the wave resistance is given by 
7/2 , 2 
1) 3 16 ex | {M2 e—2rof sec? 6 4 M’2 e~2kof sec? @ 
0 
ap ZN Gy DEO Gos A Cos sxe OO, (la) 
with 
Ky =9/u?; A=, (h—W) sec8; B=«,(k —K)sin 0 sec? 0. 
This can easily be extended to continuous distributions. For distributions 
* * Proc. Roy. Soc.,’ A, vol. 118, p. 32 (1928). 
