An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

118

ROLLING SPHERE

Also if u and v are the velocities of G in the directions (1)
and (2), by Art. 115
f, =i-v0; f2= + u6.
And, since there is no slipping, the point P of the sphere has
the same velocity as the point P of the plane, whence
aW 2- == 0, i.e. ao2= U,
and aw1 + v = xQ, i.e. aw1 = x~ - v,
where f2 is the constant angular velocity of the plane.
Substituting in equations (i) and (ii) we obtain
7 j + U = 2xt2,
and 7iu-7v6= -2xQ06,
showing that f2 = 2 f2,
and fi= - xQ20.
Taking r, 0 as the polar coordinates of the point P, these
equations may be written
dt(r............................(iii)
and r-r 2= - f r........................ (iv)
Integrating (iii) we obtain
r20 = r2 +a constant.
Since 0=0 when r=0 the constant is zero and
0 --
0-7.
Substituting in (iv) r + - r= 0.
Since dt~, the above may be written
d2r
+r = O,
whence the path of P in space is given by
r =c cos (0 + a),
where c and a are constants, showing that it is a circle.
Taking the sum of the squares of r and r0 derived from this
equation, we see that the velocity is constant and equal to that
of projection.