254 ON A DYNAMICAL TOP. 



on the sphere in fig. 5, we shall see that the areas described by that point, 

 if projected on the plane of yz, are swept out at the rate 



ta, 



a' 



Now the semi-axes of the projection of the spherical ellipse described by 

 the pole are 



Dividing the area of this ellipse by the area described during one revo- 

 lution of the body, we find the number of revolutions of the body during 

 the description of the ellipse 



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 revolutions 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 identical 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 x, y and z 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 there- 

 fore the areas in the ellipses above mentioned are not described 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 b = c, then 

 the angular velocity <a, 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 1 

 &-<' 



