Wave Resistance: Cases of Three-dimensional Fluid Motion. 365 
reverting to axes with the origin in the free surface. It should be noticed 
that, as in § 5, the surface pressure (41) does not give the same surface 
elevation as the moving spheroid; the surface condition in the latter case is 
that the pressure should be constant at the free surface. But (41) does give 
the same wave formation as the spheroid, and that is the part of the surface 
effect upon which the wave resistance depends. The complete surface eleva- 
tion can be easily written down by direct methods as in the case of the 
submerged sphere. 
Using (41) now as an example of (35), we find the wave resistance of the 
spheroid from (37) ; after integrating with respect to / and &, the result is 
ar | 2 
Re 128 gpa | sec? he-2oF8"4 {Tso (xyuesec p)}2deb. (42) 
0 
It can be verified that this gives the result for the sphere by making e zero. 
For a given relation between f and ae, the value of R can be obtained 
approximately by the numerical methods used in the previous examples ; 
judging from rough calculations, it appears that the resistance curve does not 
show prominent humps and hollows. This might be anticipated from the 
surface pressure (41), which can be evaluated in simple form; if we repre- 
sent the pressure distribution by a surface with p, a, and y as co-ordinates, 
then (41) gives a single oval-shaped peak with its longer axis in the direction 
Oz. On the other hand, the pressure distribution (28) represents two 
distinct peaks. We may compare in this respect the behaviour of ships’ 
models; it depends upon the shape of bow and stern, and the relation 
between them, whether the resistance curve has marked interference effects 
or is a continuously ascending curve. 
12. We have limited the previous cases to combinations of simple pressure 
systems ranged along the axis Oz. The method can obviously be extended 
to systems with their centres on Oy; or again, for systems situated in the 
plane zy, a four-fold summation in the manner of (36) would give further 
generality. For the corresponding problem of the motion of a submerged 
body, one could obtain the wave resistance of any body whose surface is 
formed of stream lines due to the combination of sources and sinks with a 
uniform stream. 
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