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 definition, 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 problem 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 1 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_ measures 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 minimum. These two are at right angles, and the third axis is perpendicular to their plane, and is called the mean axis. * Let $A$, $B$, $C$ be the moments of inertia about the principal axes through the centre of gravity, taken in order of magnitude, and let $\omega_1$ $\omega_2$ $\omega_3$ be the angular velocities about them, then the angular momenta will be $A\omega_1$, $B\omega_2$, and $C\omega_3$.