PROFESSOR TAIT ON KNOTS. 505 



great importance is this consideration, I have appended to Plate LXXIX. the 

 five figures N ; with the nature of each crossing indicated. The numbers affixed 

 show the positions they occupied in the census of 8 folds, when the crossings 

 were alternately over and under. Then they were all unique knots, incapable 

 of any change of form. Noiv they are capable of being changed into one 

 another. The linked trefoils in N, xiv. are perversions of one another. But we 

 may have them of the same kind, and the link such that there shall be 

 continuations of sign. This was briefly treated in Part I. § 42, 1. How many 

 new types may by this process be added to the census, I have not yet made 

 out with certainty even for the 8 folds. 



P.S. — I may introduce here, as a note on Part I. of this series of papers, a 

 remark or two with reference to the three-ply plaits treated there ; in § 27 as 

 fully knotted, and in § 42, 1 as fully beknotted. First, it is obvious that the 4 

 fold, as first drawn in § 17, should have been repeated in Plate XV., at the head 

 of the series of figures 15, 16, 17, &c. It is the case of 3n + 1 of § 27, with »=1. 

 Secondly, with its crossings arranged as in fig. P, Plate LXXIX. of the 

 present paper, it should have come in before figs. 24 and 25 of Plate XVI 

 Part I., in a form reducible to the ordinary trefoil. Fig. 25 of that Plate 

 puzzled me much at the time when I drew it, for I could not account for the 

 production of a 3 fold and a 5 fold (linked) from a figure possessing a peculiar 

 kind of (cyclonic ?) symmetry round an axis. The figure is accurate, but I now 

 see that it gives an erroneous impression of the true nature of the knotfulness. 

 The correct idea is at once obtained from Plate LXXIX., fig. Q, of the present 

 paper. The knot is an irreducible trefoil, with a second of the same character 

 tied twice through one of its three-cornered meshes. 



{Added, September 3, 1885.) 



Three days ago I received from Mr Lockyer a copy of a most interesting 

 pamphlet " On Knots, ivith a Census for Order Ten" a reprint from the Trans 

 Connecticut Acad., vol. vii., 1885. The author, Prof. Little of the State Uni- 

 versity, Nebraska, has made an independent census of 10 fold knots ; employ- 

 ing the partition method, with some new special rules analogous to those in 

 Mr Kirkman's recent paper. So far as I can judge from a first hasty compari- 

 son of the mere number of types and forms in each class, there are important 

 discrepancies between this census and my own. One of these, at least, is due 

 to a slip on my part ; and, as my paper was not printed off when I detected 

 it, I have taken the opportunity of correcting it both in the text and in the 

 corresponding Plate. I had failed to notice that the two forms which now 

 appear under No. 109 really belong to one type. Hence I have had to 



VOL. XXXII. PART III. 4 O 



