The following additional remarks are the outcome of my study of the polyhedral data for tenfold knottiness, which I received from Mr Kirkman on the 26th of last January. My main object was, as in the first chapter of Part II., to determine the number of different types; as well as the number of essentially different forms which each type can assume, as distinguished from mere deformations due to the mode of projection.

This study has been a somewhat protracted one, in consequence (1) of the great number of tenfold knots; (2) of the very considerable number of distortions of several of the types, many of which are essentially distinct while others present themselves in pairs differing by mere reversion; and especially (3) of the fact that the polyhedral method often presents some of the distinct forms of one and the same type projected from essentially different points of view (of which, in the present case, there are sometimes twelve in all).