PROFESSOR MAXWELL ON A DYNAMICAL TOP. 563 
increases with the size of the ellipse, so that the section corresponding to ¢?=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 ¢’=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 ¢=110 to ¢=107 the sections are ellipses, ¢=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 /, m,n, about 
the axis of a is 
soem es (1 w, +mw,+nW) 
Substituting the values of w,, w,, w,, in terms of /, m, n, and taking account of 
equation (3), this expression becomes 
(—e) 
Pua 
H 
Changing the sign and putting lao we have the angular velocity of the in- 
variable axis about that of 2. 
w, e—a? 
7/2 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 @. 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 
~T 
VOL. XXI. PART IV. 
