Mr. J. Bridge on the Gyroscope. 341 
able about a diameter at right angles to the axis of the solid by 
means of pivots 4, 6! inserted into another ring ; and this second 
ring is moveable about a fixed vertical diameter c ¢! at right angles 
to bd. 
The facts to be explained are the following :-— 
1. When the solid is in rapid rotation about aa’, let a weight 
be hung to the first supporting ring at a. Then, instead of an 
accelerated motion about 60', we have a uniform motion about cc’. 
The motion of a is in a direction a right angle in advance of that 
in which the weight alone would have caused it to move. The 
rate of this precessional motion about the axis ec! does not de- 
pend on the elevation of the axis aa’, but is increased when the 
force applied is increased, or when the rate of rotation of the 
solid of revolution is diminished. 
2. When the outer ring, or the axis 0 J/, is fixed, aa! will only 
be moveable in a vertical plane. If in this case, while the solid 
is in rapid rotation, a force be applied at a, it will produce very 
nearly the same effect as if the solid had been previously at rest. 
3. If aa! be allowed to move with great freedom in one plane 
only, fixed with respect to the earth, the axis aa’ will, during 
the rotation of the solid, oscillate about the meridian line of the 
plane in a manner similar to a common pendulum, the time of 
oscillation being a minimum when the plane in which aa’ lies is 
parailel to the earth’s axis. 
The following investigation contains an approximate solution 
of these problems. 
From Earnshaw’s ‘Dynamics,’ art. 257, it follows, that if C 
be the moment of inertia about the axis of figure of a solid, and 
A that about any other principal axis, when the body is set in 
rotation with velocity » about an axis making an angle @ with 
the axis of figure, the effect of the centrifugal forces is the same 
as that of a couple, from which the angular acceleration of the 
A 
to bring the principal axis towards the position of the instanta- 
—A , 
wo, 
A 
body originally at rest would be 
w* sin @ cos 8, tending 
neous axis. If @ be small, this is nearly equal to e 
which I will call cw*6. 
The First Problem. 
Let A be any point of the horizontal great circle on a sphere 
concentric with the solid, i the extremity of the instantaneous 
axis, a the extremity of the principal axis, 
f, Wy the spherical coordinates of i. 
¢', ' the spherical coordinates of a. 
