Three years ago I called the attention of the Society to the following theorem:—

The resultant of two pure strains is a homogeneous strain which leaves three directions unchanged; and conversely.

[It will be shown below that any strain which has three real roots can also be looked on (in an infinite number of ways) as the resultant of two others which have the same property.]

As I was anxious to introduce this proposition in my advanced class, where I was not justified in employing the extremely simple quaternion proof, I gave a number of different modes of demonstration; of which the most elementary was geometrical, and was based upon the almost obvious fact that

If there be two concentric ellipsoids, determinate in form and position, one of which remains of constant magnitude, while the other may swell or contract without limit; there are three stages at which they touch one another.

[These are, of course, (1) and (2) when one is just wholly inside or just wholly outside the other (that is when their closed curves of intersection shrink into points), and (3) when their curves of intersection intersect one another. The whole matter may obviously be simplified by first inflicting a pure strain on the two ellipsoids, such as to make one of them into a sphere, next considering their conditions of touching, and finally inflicting the reciprocal strain.]

But the normal at any point of an ellipsoid is the direction into which the radius-vector of that point is turned by a pure strain; so that for any two pure strains there are three directions which they alter alike. (These form, of course, the system of conjugate diameters common to the two ellipsoids.) This is the fundamental proposition of the paper referred to, and the theorem follows from it directly.