PROFESSOR MAXWELL ON A DYNAMICAL TOP. 563 



increases with the size of the ellipse, so that the section corresponding to e 2 =107 

 would be two parallel straight lines (beyond the bounds of the figure), after which 

 the sections would be hyperbolas. 



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

 the mean axis. They are all hyperbolas, except when <? 2 =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 2 = 110 to e 2 = 107 the sections are ellipses, e 2 ~l07 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 w L , w 2 , w 3 , in terms of /, m, n, and taking account of 

 equation (3), this expression becomes 



H x p I 



Changing the sign and putting ^=~2~jj we have the angular velocity of the in- 

 variable axis about that of x. 



l-l* 



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 2 . 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 on the 

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

 on the plane of y z, are swept out at the rate 



VOL. XXI. PART IV. 



