276 T. H. Havelock. 
with semi-axis a = 46/5; and prolate spheroids with a = 56/4, 5b/2 and 5b 
respectively. Graphs are given for the variation of wave resistance with 
velocity for these five models (i) when moving in the direction of the axis of 
revolution, (li) when moving at right angles to that axis; these illustrate 
respectively the effect of increased length, and the effect of increased beam and 
area of cross-section. It is of interest to note that increase of length gives 
diminished resistance at low speeds, with a subsequent rapid increase ; while 
increasing beam in the second series gives increased resistance at all speeds. 
2. Consider the motion of a solid bounded by the. ellipsoid 
fit ee i, (1) 
in an infinite liquid, the velocity being u parallel to Oz. 
It is well known that if V is the gravitational potential of a uniform solid of 
unit density bounded by (1), then the velocity potential of the fluid motion is 
given by 
= u OV 
$ oreo = a are (2) 
where 
ie re du 
: = abe | sy eee (3) 
eince 
dx’ dy’ dz 
ve y hs 
IN SOR POT rr Gao () 
taken throughout the ellipsoid, it follows from (2) and (4) that the velocity 
potential of the fluid motion is that due to a uniform volume distribution of 
doublets throughout the ellipsoid, with their axes parallel to Ox, and of moment 
per unit volume equal to u/27(2 — a»). 
Sunilarly for motion parallel to Oy or Oz we have a like result with a corre- 
sponding quantity 6, or y taking the place of «). For motion in any other 
direction we resolve the velocity along the three axes and combine the 
component doublet systems. 
The gravitational potentials of two solid homogeneous ellipsoids, bounded 
by confocals, at any poimt external to both are proportional to their masses. 
Hence in the hydrodynamical problem we may replace the distribution of 
doublets throughout the ellipsoid (1) by a uniform distribution through any 
interior confocal, increasing the moment per unit volume by the factor 
abe/»/{(a° + 2) (6? + 2) (2 + d)}, (5) 
where 2 is the parameter of the confocal. 
313 
