PROFESSOR TAIT ON KNOTS. 335 



The analysis of self-locked knots, such as iv. and vit. of the 8-folds, and n., ix., 

 x., xix., &c, of the 9-folds, is considered below. 



II. Beknottedness. 



7. The question of Beknottedness (on which I have occasionally made short 

 communications to the Society since my papers of 1876-7 were printed in a 

 brief condensed form) has been again forcibly impressed on me while 

 endeavouring to recognise identities among Kirkman's groups. I still con- 

 sider that its proper measure is the smallest number of changes of sign which 

 will remove all knottiness. But, shortly after my former paper was published, I 

 was led to modify some ideas on the subject, which were at least partially 

 given there. I had been so much impressed by the very singular fact of the 

 existence of amphicheiral forms, that I fancied their properties might in great 

 measure explain the inherent difficulties of this part of the subject. I have 

 since come to see that this notion was to some extent based on an imperfect 

 analogy, due to the properties of the 4-fold amphicheiral, and that the true 

 difficulty is connected with Locking. 



8. The existence and nature of this third method of entangling cords were 

 first made clear to me by one of the random, sketches which I drew to 

 illustrate Sir W. Thomson's paper on Vorte.x- Motion [Trans. R. S. E., 1867-8]. 

 I had not then even imagined that the crossings in any knot or linkage could 

 always be taken alternately over and under, though I found that I could make 

 them so in all these sketches. The particular figure above referred to again 

 presented itself, among others possessing a similar character, while I was 

 studying the peculiar group of plaited knots whose schemes contain the lettering 

 n alphabetical order in the even as well as in the odd places. (See §§ 27, 42, 

 of my former paper.) But I soon saw that, though I had first detected locking 

 in those members of the group of plaits where three separate strings are 

 involved, essentially the same sort of thing occurs in the other members of the 

 group, though they are also proper knots in the sense of being each formed 



with a single continuous and endless string. And, as the above very simple 

 example sufficiently shows, we can have locking, independent of either knotting 

 or linking, with two separate strings. For it is clear that the irreducibility 



VOL. XXXII. PART II. 3 I 



