ON A DYNAMICAL TOP. 253 



*The second figure represents the sections made by a plane, perpendicular 

 to the mean axis. They are all hyperbolas, except when e 3 =107, when the 

 section is two intersecting straight lines. 



The third figure shows the sections perpendicular to the axis of least 

 moment of inertia. From e'=110 to e 2 =107 the sections are ellipses, e 2 =107 

 gives two parallel straight lines, and beyond these the curves are hyperbolas. 



*The fourth and fifth figures show the sections of the series of cones 



made by a cube and a sphere respectively. The use of these figures is to 



exhibit the connexion between the different curves described about the three 

 principal axes by the invariable axis during the motion of the body. 



*We have next to compare the velocity of the invariable axis with respect 

 to the body, with that of the body itself round one of the principal axes. 

 Since the invariable axis is fixed in space, its motion relative to the body 

 must be equal and opposite to that of the portion of the body through which 

 it passes. Now the angular velocity of a portion of the body whose direction - 

 cosines are I, m, n, about the axis of x is 



Substituting the values of <a,, a>,, <u,, in terms of I, m, n, and taking 

 account of equation (3), this expression becomes 



1-P 

 Changing the sign and putting I = -jp. we have the angular velocity of 



Ct l 



the invariable axis about that of x 



a* 



always positive about the axis of greatest moment, negative about that of least 

 moment, and positive or negative about the mean axis according to the value 

 of e*. The direction of the motion in every case is represented by the arrows 

 in the figures. The arrows on the outside of each figure indicate the direction 

 of rotation of the body. 



*If we attend to the curve described by the pole of the invariable axis 



