504 PROFESSOR TAIT ON KNOTS. 



22. The nature of the special difficulty hinted at in the beginning of the 

 paper will be easily seen from the simple case illustrated by the four figures M, 

 Plate LXXIX. They denote various forms of the type 40 of Plate LXXX. 



It will be noticed that the crossings A, B, C may, one, two, or all, be 

 changed from one lap of the string to the other, as shown in the second figure. 

 Also D may be transferred to a position between A and B, or between A and C. 

 There are thus two positions for each of A, B, and C ; and three positions for 

 D ; giving 24 combinations in all. But it is clear that we need not shift D at 

 all, so far as the outline of the figure is concerned; for a mere rotation of the 

 whole in its own plane (as A, B, and C are similar to one another) will effect 

 this. Then a change of B will merely give the reverse of the figure obtained by 

 changing C. Again, by inverting the first figure about a point in the inner 

 mesh, we get the second. If we had changed C, and then inverted, we should 

 have got the same figure as by changing simultaneously A and B. By changing 

 C alone in the first, Ave get the third ; but by shifting D in the first we get the 

 fourth ; and these two are obviously each the reverse of the other. Thus the 

 24 figures reduce to the three shown in Plate LXXX. As another example, 

 take the third form of the third type of 10 folds as given in Plate LXXX. 

 Two of thecrossings on its external boundary can be shifted, but each to one 

 other place only. The form itself, and the same with one or both of these 

 crossings shifted, give a set of four ; each of which can take five new forms by 

 the shifting of other crossings. But it will be found that the 24 forms thus 

 obtained are identical in pairs ; — thus reducing to the 12 given in the Plate. 



23. Mr Kikkman informs me that he has nearly completed the enumeration 

 and description of the polyhedra corresponding to the unifilar 11 folds. I 

 hope, therefore, at some future time to lay before the Society the census of 

 11 fold knottiness. This was the limit to which I ventured to aspire nearly 

 two years ago, in a paper" which, I am happy to think, directed Mr Kirkman's 

 attention to the subject. 



24. It must be remembered that, so far as these instalments of the census 

 have gone, we have proceeded on the supposition that in each form the crossings 

 have been taken over and under alternately. But, as was shown in § 13 of 

 Part I., as soon as we come to 8 folds we have some knots which may preserve 

 their knottiness even when this condition is not fulfilled. These ought, there- 

 fore, to be regarded as proper knots and to be included in the census as new 

 and distinct types. This is a difficulty of a very formidable order. It depends 

 upon the property which I have called Knotfulness (Part I. § 35 ; II. § 6), 

 for whose treatment I have not yet managed to devise any but tentative 

 methods. 



To show, by a single case (even though not thoroughly worked out) of how 



* Listing's Topologie, §22, Phil. Mug., Jan. 1884. 



