564 PROFESSOR MAXWELL ON A DYNAMICAL TOP. 
Now the axes of the projection of the spherical ellipse described by the pole are, 
/ era and e—a? 
NB —a2 C_—ae 
Dividing the area of this ellipse by the area described during one revolution 
of the body, we find the number of revolutions of the body during the description 
of the ellipse— 
a? 
VR Re Vee 
The projections of the spherical ellipses upon the plane of yz are all similar 
ellipses, and described in the same number of revolutions; and in each ellipse so 
projected, the area described in any time is proportional to the number of revo- 
lutions of the body about the axis of x, so that if we measure time by revolutions 
of the body, the motion of the projection of the pole of the invariable axis is iden- 
tical with that of a body acted on by an attractive central force varying directly 
as the distance. In the case of the hyperbolas in the plane of the greatest and 
least axis, this force must be supposed repulsive. The dots in the figures 1, 2, 3, 
are intended to indicate roughly the progress made by the invariable axis during 
each revolution of the body about the axis of 2, 7, and 2 respectively. It must 
be remembered, that the rotation about these axes varies with their inclination to 
the invariable axis, so that the angular velocity diminishes as the inclination 
increases, and therefore the areas in the ellipses above mentioned are not de- 
scribed with uniform velocity in absolute time, but are less rapidly swept out at 
the extremities of the major axis than at those of the minor. 
* When two of the axes have equal moments of inertia, or ) = ¢, then the 
angular velocity “, is constant, and the path of the invariable axis is circular, 
the number of revolutions of the body during one circuit of the invariable axis, 
being 
a? 
b? a? 
The motion is in the same direction as that of rotation, or in the opposite 
direction, according as the axis of w is that of greatest or of least moment of 
inertia. 
* Both in this case, and in that in which the three axes are unequal, the 
motion of the invariable axis in the body may be rendered very slow by diminish- 
ing the difference of the moments of inertia. The angular velocity of the axis 
of « about the invariable axis in space is 
et —a?E 
“a(1—P) 
which is greater or less than ,, as ¢ is greater or less than a’, and, when these 
quantities are nearly equal, is very nearly the same as %, itself. This quantity 
indicates the rate of revolution of the axle of the top about its mean position, 
and is very easily observed. 
