Wave Resistance of a Spheroid. 283 
has been found simpler to make numerical calculations directly from the 
integrals as given, although a considerable amount of work is involved in any 
case. 
The calculations have been carried out for a set of five spheroids, including 
the sphere, the radius 6 of the central circular section being supposed constant 
and the semi-axis a@ varied. The following are the data for the series :—A, 
oblate, a = 4/5, e’ = 0-6; B, sphere, a = 0; C, prolate, a = 5b/4,e = 0-6; 
D, prolate, a = 5b/2, e = 0-9165; E, prolate, a = 5b, e = 0-9798. The axial 
sections of these forms are shown in fig. 1, drawn to scale, the diagram 
showing one quarter of the section in each case. 
We suppose the axis horizontal in each case and at the same depth f below 
the free surface. To make a definite case for numerical calculation we take 
f = 2, (42) 
that is, the depth twice the radius of the central circular section. We consider 
the models in two series, (i) with the axis in the direction of motion, (ii) with 
the axis at right angles to the motion. Our object is to show the variation of 
wave-resistance with velocity for each model, and to see how the graph varies, 
in (i) with increasing length, and in (ii) with increasing beam. To give one 
example of the calculations, when a = 5b/2, (27) gives 
R= 2240ngpb%e~? |. cP LJ, 9(0-5728 pNi+e?)}? dt. (48) 
For velocities which are of special interest, the parameter p ranges from about 
1 to 8. A graph of the Bessel function Js. was drawn on a large scale and 
values were taken from it, except for small values of the argument when they 
were calculated from tables of J). and J_,/. Values of the integrand were 
calculated for values .of ¢ at intervals of 0-1, and the numerical integration 
carried out by the usual methods. Owing to the exponential factor, it was 
unnecessary to go beyond ¢ = 2 in any case; and for the larger values of p, 
a smaller range of ¢ was sufficient. This process was carried out for seven or 
eight values of p, and so a graph could be drawn for the variation of R with p, 
that is, with velocity wu. 
320 
