Wave Resistance: Effect of Parallet Middle Body. 91 
The roots correspond to the series of humps and hollows on the resistance 
curve. The third row shows the wave-making length Z, and in the last row 
are the values of V/\/L, where V is in knots and L is the length of the ship 
in feet. The velocities which would be assigned from an inspection of the 
actual resistance curve would naturally be a little higher than those found 
from (34), especially where the mean resistance curve rises rapidly. We have 
already considered the increase of Z from moderate to higher velocities ; we 
notice here that it is not sufficient to affect appreciably the value of V/,/L for 
the position of the final hump. Table IV. brings out a new point, namely, 
the increase of Z with decreasing velocity. It is easy to see how this arises. 
We may express it in this way: the particular model has straight lines at bow 
and stern, including a finite angle, and as the velocity decreases there is an 
increase in the relative importance of the wave-making properties of the ends 
compared with the parts where the change of curvature is gradual ; or, analy- 
tically from (34), when « is large we can use the asymptotic values of the P 
functions, and the roots approximate to those of P, (2x) = 0 and succeed 
each other at intervals of 7/2. It has not been found possible to analyse 
experimental curves to see if this effect occurs ; the interference at low velo- 
cities is small and unimportant in practice, and the curves are not sufficiently 
accurate for the purpose. One reference may, however, be given where this 
effect seems to have been observed. 
In a contribution to a recent discussion quoted in §7, G. Kempf describes 
some experiments made at the Hamburg Experimental Tank. The model 
was of cylindrical form with a hemispherical entrance and a run formed by 
the rotation of a sine curve ; it is stated that Z was not constant at all speeds, 
but that the value of ,/Z increased 10 per cent. with decreasing speed from 
V, to V;. It may be noted, as a coincidence, that in Table IV., Z increases 
from 126 at V, to 145 at V;, and this is an increase of 7 per cent. in ,/Z. 
To show the effect of parallel middle body, we consider finally a ship of 
400 ft., with the same entrance and run as before, but with 240 ft. of parallel 
middle body. 
Since y = 3z, equation (33) becomes in this case 
=o 
1 
2x" 
3 1 1 1 1 
|= bot et Py ++ = Ps) {5a} (Lp.— =Pa) {4x}-+ ames {3a} 10: 
(35) 
Table V gives the roots and the similar quantities deduced from them, 
228 
