78 T. H. Havelock. 
a ship with entrance and run each of 80 feet and with parallel middle body 
increased from zero up to 340 feet, as in W. Froude’s well-known experiments 
($6). Fig. 2 shows the curves of wave resistance for five different velocities, 
the base being the length of parallel middle body. A short account of model 
results and of recent discussions is given in §7, and the present calculations are 
reviewed in the remaining sections. The information from the curves of fig. 2 
is extended by an equation whose roots give the complete series of maxima 
and minima (§8). The roots are found numerically for three series. With both 
the length of the ship and the speed varying, we obtain the roots for the maxima 
for which on a simple theory the wave-making length is equal to one and a half 
wave-lengths ; Table IIT shows the actual variation of this length. Then two 
series of roots are found for a ship of constant length at varying velocities, 
one for a ship of 160 feet without parallel middle body, and the other when 
240 feet of parallel middle body have been inserted; these are given with 
other quantities in Tables IV and V, and the results are discussed in relation to 
experimental data. 
Expressions for Wave Resistance. 
2. A uniform stream of deep water moves with velocity c in the negative 
direction of Oz, the axes Ox, Oy being in the undisturbed surface, and the 
axis Oz vertically upwards. Suppose there is a doublet of moment M in the 
liquid at the point (h, 0, —f) with its axis parallel to Oz. With the usual 
limitation of assuming the additional fluid velocity at the surface to be small 
compared with c, one can write down complete expressions for the velocity 
potential, and so deduce the wave disturbance and the corresponding wave 
resistance. It is convenient to begin here by quoting from a previous paper* 
the wave resistance, altered to the present notation, as 
wla 
i LérgteMe-* | sec® he~ Cul sectd qd, (1) 
0 
In the same paper it was also shown how to generalize this expression, 
first, for any two doublets at given points in the plane y = 0 and then for any 
continuous distribution in the same plane. Equation (37) of that paper 
gives the result for a continuous line distribution of doublets along the line 
y =0, = —f; an obvious extension gives now 
enone jale i. dh | ae (i bi NYS) 
x secd few Ft Nietsec?# cos [{g (h—h')/c?} sec 4] dd, (2) 
* ©Roy. Soc. Proc.,’ A, vol. 95, p. 358 (1919). 
