13·1. If the mass of every element of a body or particle of a system be multiplied by the square of its distance from an axis, the sum of the products is called the Moment of Inertia of the body or system about that axis. Thus if m denotes the mass of an element or particle and r denotes its distance from the axis, the moment of inertia is Σmr2.

In like manner we may define the moment of inertia of a system with respect to a point or plane as Σmr2, where r denotes distance from the point or plane.

If M denotes the whole mass and κ be a line of such length that Mk2 is the moment of inertia about an axis, then κ is called the radius of gyration of the system about that axis.

When rectangular coordinate axes are used, the moments of inertia of a body about the axes are denoted by

A = Σm(y2 + z2), B = Σm(z2 + x2), C = Σm(x2 + y2);

and the sums represented by

D = Σmyz, E = Σmzx, F= Σmxy

are called the products of inertia of the body with regard to the axes yz, zx, xy.

13·1. Theorem of Parallel Axes.The moment of inertia of a body about any axis is equal to its moment of inertia about a parallel axis through its centre of gravity, together with the product of the whole mass and the square of the distance between the axes.

Let the given axis be taken as axis of z.