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 Hayward, in his paper already referred to, has made great use of the 

 mechanical conception of Angular Momentum. 



Definition. — 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- 



