155 
1914-15.] The Theory of the Gyroscope. 
2. In cases of “ precessional ” motion, the pole C describes a circle on 
the unit sphere with constant angular velocity If 0 be the angular 
radius of this circle, the acceleration of C towards its centre is t;Vsin 0, and 
the component at right angles to the path, in the tangent plane, is therefore 
cot 0. Hence, if there are no external forces, 
cot 0 = Cnv, 
or, since v — sin 0 v/y , 
i/r = Cn/(A cos 0) . . . . . • (6) 
which is the ordinary formula for the free “ Eulerian nutation.” 
The same formula applies to the ‘‘rapid” precession of a top whose 
velocity of spin is very great, gravity being in this case relatively unim- 
portant. 
In the case of the “ slow ” precession we may ignore the acceleration, 
which involves the square of v, and equate the deviating force Cnv, or 
Cn sin 0 to the effective component of gravity, viz. ALgh sin 0 in the 
ordinary notation. Thus 
ij/ = Mgh/Cn ...... (7) 
The exact condition for steady motion, including both cases, is obviously 
Av^cot^= - sin ^ -H Cwv . . . . • (S) 
or 
A sin ^ cos ^ = - Mp'/i sin ^ -1 - Ctz sin ^ . . . • (9) 
The small oscillations of a rapidly spinning top about a state of precessional 
motion are also easily investigated. Suppose, for instance, that the pole C 
is initially at rest. It will at once begin to descend, but the deviating force 
which is quickly called into play will deflect it continually to the left, so 
that it presently turns upwards again, describing a sort of cycloidal curve. 
When the undulations are small the circumstances are very closely analogous 
to the case of a particle moving in a plane under two forces, one of 
which is constant in magnitude and direction, whilst the other is at right 
angles to the path and varies as the velocity.* The equations of motion 
in such a case are of the forms 
i/=f+/3x (10) 
whence 
x = ct + a sin /3t, y — a cos (St . . . . (11) 
if the origins of x, y, t be suitably adjusted. The path is therefore a 
trochoid, the period of oscillation about the uniform rectilinear motion being 
* This case occurs in Hydrodynamics, in the motion of a cylinder with cyclic irrotational 
motion about it, and subject to a constant force such as gra\dty. Again, in Electricity we 
have the case of an electron moving in a field where the electric and magnetic forces are 
uniform and at right angles to one another. 
