134 T. H. HAVELOCK 
For G, we expand the corresponding term ¢; in spherical harmonics, and 
we find that G,, reduces to the same expression (21) with a negative sign; 
thus the total moment is zero, as should be the case. Turning to the 
expressions in (9) and (19) for the prolate spheroid, we find their limiting 
values for e > 0 reduce to the correct values for the sphere. 
6. Returning to the spheroid, we notice that G, may be positive or 
negative according to the speed; on the other hand, G is essentially 
negative. Further, from (20) we have 
G, = —BR/K, A. (22) 
If k, and k, are the inertia coefficients for axial motion and transverse 
motion respectively, we have 2e3A = (1—e?)(1++k,) and a similar relation 
between B and k,. Hence (22) may be put in the convenient form 
Gy = —(1+k)R/xo(1+hy). (23) 
The ratio (1+-4,)/(1+4,) is unity for a sphere, and approximates to two 
for a spheroid of large length-beam ratio. When c—> oo, or x)— 0, the 
integrals in the expressions for Rf, G, and G, all reduce to the integral 
given in (21), which can be expressed in terms of Bessel functions; hence 
we may find the limiting values of these quantities as the speed increases 
indefinitely. It appears that as c—> oo, R becomes zero of order c-?; on the 
other hand G, and G, approach finite limiting values, with 
G,> ngpaeA2/f?, Gy > —Eaxgpa%e®A B/f?. (24) 
Thus the moment G approaches the limiting value 
G = G+ G,—> —gngpa*b*(1--ky)(ky—ky)/f?, (25) 
and this is negative for a prolate spheroid. 
Some numerical values have been calculated from (20) for a spheroid of 
a length-beam ratio of 10. The moment at low speeds may be positive or 
negative and is small numerically; after a Froude number, c/,/(2ga), of 
more than about 0-4 the moment remains negative and increases rapidly 
towards its limiting value. 
It may seem unexpected, as compared with surface ships, to find the 
moment remaining negative at high speeds. The model of a surface ship 
is usually allowed to trim and at high speeds it takes up a position with 
bow up and stern down, corresponding to a positive moment; the attitude 
of the model is then roughly parallel to the mean line of the water surface 
in its vicinity. But the submerged spheroid we have been considering has 
its axis maintained horizontal; so we may describe it roughly as being in 
a stream whose effective direction in the vicinity of the spheroid is inclined 
to the axis and this provides a moment tending to increase this angle, that 
is, a negative moment. For a numerical case take a spheroid with a = 106 
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