PROFESSOR MAXWELL ON A DYNAMICAL TOP. 561 
manence of the original angular momentum in direction and magnitude, and the 
permanence of the original vis viva. 
* The mathematical difficulties of the theory of rotation arise chiefly from the 
want of geometrical illustrations and sensible images, by which we might fix the 
results of analysis in our minds. 
It is easy to understand the motion of a body revolving about a fixed axle. 
Every point in the body describes a circle about the axis, and returns to its 
original position after each complete revolution. But if the axle itself be in 
motion, the paths of the different points of the body will no longer be circular or 
re-entrant. Even the velocity of rotation about the axis requires a careful defi- 
nition, and the proposition that, in all motion about a fixed point, there is always 
one line of particles forming an instantaneous axis, is usually given in the form 
of a very repulsive mass of calculation. Most of these difficulties may be got 
rid of by devoting a little attention to the mechanics and geometry of the pro- 
blem before entering on the discussion of the equations. 
Mr Haywarp, in his paper already referred to, has made great use of the 
mechanical conception of Angular Momentum. 
Derinition.— The Angular Momentum of a particle about an axis is measured 
by the product of the mass of the particle, its velocity resolved in the normal plane, 
and the perpendicular from the axis on the direction of motion. 
* The angular momentum of any system about an axis is the algebraical sum 
of the angular momenta of its parts. 
As the rate of change of the linear momentum of a particle measures the 
moving force which acts on it, so the rate of change of angular momentum mea- 
sures the moment of that force about an axis. . 
All actions between the parts of a system, being pairs of equal and opposite 
forces, produce equal and opposite changes in the angular momentum of those 
parts. Hence the whole angular momentum of the system is not affected by 
these actions and re-actions. 
* When a system of invariable form revolves about an axis, the angular 
velocity of every part is the same, and the angular momentum about the axis 
is the product of the angular velocity and the moment of inertia about that 
axis. 
* It is only in particular cases, however, that the whole angular momentum 
can be estimated in this way. In general, the axis of angular momentum differs 
from the axis of rotation, so that there will be a residual angular momentum 
about an axis perpendicular to that of rotation, unless that axis has one of three 
positions, called the principal axes of the body. 
By referring everything to these three axes, the theory is greatly simplified. 
The moment of inertia about one of these axes is greater than that about any 
other axis through the same point, and that about one of the others is a mini- 
