ON THE ROTATION REQUIRED FOR THE STABILITY OF AN 
ELONGATED PROJECTILE. 
BY 
A. G. GREEN HILL, M.A. 
(Professor of Mathematics to the Advanced Class of Artillery Officers.) 
Suppose an elongated projectile—for example, in the shape of a 
prolate spheroid—moving in infinite frictionless homogeneous liquid 
of density p under no forces. 
Let 0 be the centre of the spheroid, OC the axis of figure, OA and 
OB two equatoreal radii at right angles to one another fixed in the 
body; and let u, v, w be the component linear velocities of the body 
parallel to OA, OB, and OC; and p, y, r the component angular velo¬ 
cities of the solid about the axes OA, OB, and OC. 
If the velocity u only had existed, then the kinetic energy of the 
body and the surrounding liquid would be \c 11 u 9, ; where c u is a 
constant depending on the mass and shape of the body and the density 
of the liquid; c u is the force which would have to act on the body 
for the unit of time to produce the unit of velocity parallel to OA, 
and is called the effective inertia of the body in the liquid in the 
direction OA; therefore, also, C\\U in this case is the resultant linear 
momentum of the solid and liquid. 
If the density of the surrounding liquid had been zero, then c n 
would be equal to the number of units of mass M in the body; but in 
consequence of the velocity u of the body generating a certain state of 
motion in the surrounding liquid, we shall have 
Cn =if(l +Fa); 
where a is a certain constant depending on the shape of the body, 
o- being the density of the body. 
Since the body is of revolution, it follows that if the velocity v only 
had existed, the kinetic energy of the system would be \c Y] v i , and the 
momentum CuV. 
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