A short paper of mine on Fagnani's theorem, and on Confocal Conic Sections, was inserted in the twenty-third volume of the Transactions of the Royal Society. Some of the conclusions of that paper can, however, be obtained more simply, as I will now proceed to show.

I will, in the first place, resume the problem—

“To find the intersection of a confocal ellipse and hyperbola.”

Since the curves have the same foci, and therefore the same centre, let the distance between the centre and focus be called unity, since it is the same for both curves. Let a, b, be the axes of the ellipse, A, B, those of the hyperbola. Then we have 1 = a2 − b2 = A2 + B2, which equation expresses the condition of confocality.