1. A year ago last April, Prof. Tait proposed that I should undertake to derive from Mr Kirkman's polyhedral drawings the alternate ± knots of eleven crossings, thus doing for order 11 what had been done so admirably by himself in orders 8, 9, and 10.

2. The work has been a very protracted one, because of the great number of forms involved—more than three times as many as in all preceding orders combined. Mr Kirkman's manuscript contains 1581 forms, of which 22 are bifilar and 16 duplicates. I find from the remainder 357 knots with 1595 forms as shown in the following table:—

As this is an odd order, perversion doubles these numbers, making 714 elevenfold knots, with crossings alternately over and under.

3. It has been thought unnecessary to show upon the Plates more than one form of each knot; all, however, have been drawn. Knots of each class having the same number of forms are grouped together to make more simple the identification of a particular elevenfold. A small figure following the series number upon the plates indicates how many distinct forms each knot can assume. Knots 84, 357, and 2386 are misplaced.