562 PROFESSOR MAXWELL ON A DYNAMICAL TOP. 
mum. These two are at right angles, and the third axis is perpendicular to 
their plan, and is called the mean axis. 
* Let A, B, C be the moments of inertia about the principal axis through the 
centre of gravity, taken in order of magnitude, and let #, w, w, be the angular 
velocities about them, then the angular momentum will be Aw,, Bw, and Cw,. 
Angular momentum may be compounded like forces or velocities, by the law of 
the “ parallelogram,” and since these three are at right angles to each other, their 
resultant is 
V A207 + Bw,” + C?w,’=H alt ts ; ; (1) 
and this must be constant, both in magnitude and direction in space, since no 
external forces act on the body. 
We shall call this axis of angular momentum the invariable awis. It is per- 
pendicular to what has been called the invariable plane. Potnsor calls it the axis 
of the couple of impulsion. The direction-cosines of this axis in the body are, 
AM Ha ae 
egy 7 Sly ORE 
Since /, m, and ” vary during the motion, we need some additional condition 
to determine the relation between them. We find this in the property of the vis- 
viva of a system of invariable form in which there is no friction. The vis-viva of 
such a system must be constant. We express this in the equation 
Aw,? + Bu,? + Cw,?= V; : : f (2) 
Substituting the values of w, w, w, in terms of /, m, 
2m? nV 
AV BC eri 
Let += a, a B?, = ¢, wee 
and this equation becomes 
OPP m+ eae . . . . « QB) 
and the equation to the cone, described by the invariable axis within the body, 
a (a? —e*) x? +(b? —e?) y? + (c? —e)2?=0 : ; é (4) 
The intersections of this cone with planes perpendicular to the principal axes 
are found by putting 2, y, or z, constant in this equation. By giving ¢ various 
values, all the different paths of the pole of the invariable axis, corresponding to 
different initial circumstances, may be traced. 
* In the figures, I have supposed a?=100, 6°=107, and c?=110. The first 
figure represents a section of the various cones by a plane perpendicular to the 
axis of x, which is that of greatest moment of inertia. These sections are ellipses 
having their major axis parallel to the axis of 6. The value of ¢ corresponding 
to each of these curves is indicated by figures beside the curve. The ellipticity 
