Rather more than twenty years ago, in a note on this subject, it was shown to the Edinburgh Mathematical Society (Proceedings, II., pp. 16–18) that a special form of continuant, viz., one with univarial diagonals, could be expressed by means of a similar continuant of much lower order. A new mode of proving this theorem, which has lately been hit upon, has unexpectedly led to the discovery that the peculiarity in question is not confined to this special form, but characterises continuants of any form whatever.