119 
ON THE ROTATION REQUIRED FOR THE STABILITY 
OF AN ELONGATED PROJECTILE. 
(AN ELEMENTARY DEMONSTRATION.) 
BY 
A. G. GKEENHILL, M.A. 
(Professor of Mathematics to the Advanced Class of Artillery Officers.) 
When a body moves in a medium it sets the medium in motion, and 
the inertia of the body—that is, its resistance to change of motion— 
is no longer necessarily the same in all directions, as it would be in a 
vacuum. 
Consider an elongated projectile of revolution moving in air under 
no forces, and let c 1 denote the inertia of the body to motion perpen¬ 
dicular to its principal axis, c 3 the inertia of the body to motion in the 
direction of that axis ; then if u, w be the component velocities per¬ 
pendicular to and in the direction of the axis, c Y u and c z w will be the 
components of linear momentum in those directions respectively; 
and if no forces act on the body, cfti and c%w will have a resultant, 
Z suppose, fixed in magnitude and direction, by the principle of the 
conservation of linear momentum. 
If 0 be the centre of the body, and if p be the component angular 
velocity about an axis OA, perpendicular to the axis of figure, then 
this motion of the body will stir up the surrounding medium; and if 
cj) be the component angular momentum about OA of the body and 
medium, then is called the effective moment of inertia of the body 
about an equatoreal axis. 
If r be the component angular velocity about the axis OC of figure, 
then, since this angular velocity will not stir up the surrounding medium, 
the body being supposed to be a smooth solid of revolution, c 6 r will be 
the component angular momentum about OC, c 6 being the moment of 
inertia of the body about OC; r will remain constant during the 
motion, the body being smooth and the medium frictionless. 
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