In the following Abstract I refer to such Linear and Vector Functions, only, as correspond to homogeneous strains which a piece of actual matter can undergo. There is no difficulty:—though caution is often called for:—in extending the propositions to cases which are not realizable in physics.

The inquiry arose from a desire to ascertain the exact nature of the strain when, though it is not pure, the roots of its cubic are all real:—i.e. when three lines of particles, not originally at right angles to one another, are left by it unchanged in direction.

1. The sum, and the product (or the quotient), of two linear and vector functions are also linear and vector functions. But, while the sum is always self-conjugate if the separate functions are so (or if they be conjugate to one another), the product (or quotient) is in general not self-conjugate:—though the determining cubic has, in this case, real roots. The proof can be given in many simple forms.