104 
ROTATION OP PROJECTILES. 
ellipsoid* (of winch the spheroid is a particular case) moving in a 
frictionless and incompressible medium. 
The prolate spheroid is the form assumed in the present instance, as 
it most nearly resembles that of an ordinary elongated projectile. 
Throughout the paper the centre of gravity is supposed to be coin¬ 
cident with the centre of figure— i.e., in the centre of the longer axis. 
Secondly, as to the problem to be investigated. If a projectile left 
the bore of a gun perfectly truly centred, and perfectly true in form, 
then, under the assumptions described above , it would require no spin at all. 
If, however, gravity be taken into account—as must necessarily be 
the case in practice—it is absolutely necessary to give spin, even in 
the case of the other assumed conditions remaining unaltered. The 
axis of the shot being, immediately after it leaves the bore, inclined to 
the tangent to the trajectory, the shot would turn over were it not 
kept in stability by its rotation. In an analogous manner, a perfectly- 
formed top ought theoretically to stand on its point without being 
spun. Practically we cannot make a top do so. 
Supposing it to be possible to succeed in making the top stand on 
its point, or the projectile fly properly through the medium, without 
giving them spin, they would obviously be in a condition of highly 
unstable equilibrium. We have, then, to find out how much spin it is 
necessary to give the projectile to convert this unstable into stable 
equilibrium. 
If we spin a top with considerable angular velocity on a table, we 
find that its longer axis is at first, as a rule, by no means upright, but 
that this axis will describe a series of cones having the point as a 
common vertex. The vertical angle of these cones becomes smaller 
and smaller till the axis of the top is upright, and the top “ sleeps/* 
This effect is produced by the friction of the point on the table. 
In a similar manner a projectile describes a helical course round the 
mean trajectory, and, our hypothesis assuming the surrounding medium 
to be frictionless, it continues to preserve this helical motion through¬ 
out its flight; though in practice the friction of the air reduces this 
motion, and tends to keep the longer axis of the projectile truly in the 
line of the mean trajectory, or (as far as the helical motion is concerned) 
to make the projectile “ sleep/* like the top. 
Eeturning to the top, which we suppose to be “ sleeping.** After a 
time, the friction of the point on the table and other causes reduce the 
angular velocity to such an extent that the top begins again to describe 
cones with its longer axis about the point as a vertex, the spin now 
being insufficient to keep that axis in stable equilibrium. The angular 
velocity which is just sufficient to keep the top “asleep** instead of 
* The ellipsoid is a solid generated by a variable ellipse which moves parallel to itself, with its 
axes in two fixed planes and with its vertices in two ellipses in those planes, having a common axis 
coincident with the intersection of the planes. The spheroid is a particular case of the ellipsoid, 
in which the variable ellipse becomes a variable circle. It may also be generated by the revolution 
of an ellipse about one of its axes. The prolate spheroid is that generated by the revolution of an 
-ellipse about its major axis. 
