Wave Resistance of an Ellipsoid. 484 
these oscillations are due to the Bessel function J in the integrards, the 
modified Bessel function I being non-oscillatory; and one might trace the 
relative importance and positions of these humps and hollows with variation 
of the quantity «?, that is, (62 — c?)/(a2 — b2). For instance, in (14) the second 
integral is non-oscillatory ; and, as one would expect, it becomes of less 
relative importance as the ratio of a to b is increased. Or, again, consider the 
positions of the humps and hollows. The maxima on the resistance-velocity 
curve will be in the neighbourhood of the maxima and minima of 
J 3/2 (KoV a? = ()), 
while the minima will be near the zeros of this function. Suppose, as an example, 
we take a = 5b and compare ellipsoids with different ratios of c to b. When 
c lies between zero and 6, the factor (1 — a?) in the integrand of (14) lies 
between 1 — ,,#2 and unity; further, if in (21) we take c as much as 2b, the 
corresponding factor is 1 + 12. It is clear, without further calculation, that 
the positions of the interference maxima and minima will be altered only 
very slightly by such a variation in beam when the ratio of length to draught 
is five or more. It appears in fact, that when the beam and draught are of 
the same order of magnitude and the length is of the order of 10 times either 
of these quantities, the form of the resistance-velocity curve is comparatively 
insensitive to changes in beam. This consideration may, perhaps, account 
partly for the measure of agreement which has been obtained between calqu- 
lated values of the wave resistance of ship models and experimental results ; 
the theory, of course, fails in many details, but the agreement in general 
character is better than might have been anticipated in view of the simplify- 
ing assumptions which have to be made. 
6. The calculations for ship models are usually made from Michell’s formula 
for the wave resistance. That expression holds for a model with a longitudinal 
vertical plane of symmetry, and is derived from an assigned distribution of 
horizontal velocity at right angles to that plane ; it is, in fact, the same ag can 
be obtained from a distribution of sources and sinks, or of horizontal doublets, 
in the vertical plane. In applying the expression to a ship there are two 
approximations, which probably involve the same limitation; one is in 
extending the distribution right up to the surface of the water, and the other 
is in obtaining the equivalent distribution from the slope of the ship’s surface. 
The latter approximation could, of course, be examined quite independently 
of the wave phenomena, but it is of interest to compare the expressions for the 
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