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XIX.— On Knots. Part II By Professor Tait. (Plate XLIV.) 



(Read 2nd June 1884. ) 



One main object of the present brief paper is to take advantage of the 

 results obtained by Kirkman,* and thus to extend my census of distinct forms 

 to knottiness of the 8th and 9th orders ; for the carrying out of which, by my 

 own methods, I could not find time. But I employ the opportunity to give, in 

 a more extended form than that in the short abstract in the Proceedings, some 

 results connected with the general subject of knots, which were communi- 

 cated to the Society on January 6, 1879, as well as others communicated at a 

 later date, but not yet printed even in abstract. 



I. Census of S-Fold and of Q-Fold Knottiness. 



1. The method devised and employed by Kirkman is undoubtedly much 

 less laborious than the thoroughly exhaustive process (depending on the 

 Scheme) which was fully described and illustrated in my former paper f; but it 

 shares, with the Partition method, which I described in § 21 of that paper and 

 to which it has some resemblance, the disadvantage of being to a greater or less 

 extent tentative. Not that the rules laid down, either in Kirkman's method or 

 in my partition method, leave any room for mere guessing, but that they are too 

 complex to be always completely kept in view. Thus we cannot be absolutely 

 certain that by means of such processes we have obtained all the essentially 

 different forms which the definition we employ comprehends. This is proved 

 by the fact that, by the partition method, I detected certain omissions in 

 Kirkman's list, which in their turn enabled him to discover others, all of which 

 have now been corrected. And, on this ground, the present census may still 

 err in defect, though such an error is now perhaps not very probable. 



On the other hand, the treatment to which I have subjected Kirkman's col- 

 lection of forms, in order to group together all mere varieties or transformations 

 of one special form, is undoubtedly still more tentative in its nature ; and 

 thus, though I have grouped together many widely different but equivalent 

 forms, I cannot be absolutely certain that all those groups are essentially 

 different one from another. • 



Unfortunately these sources of possible error, though they tend (numeri- 

 cally) in opposite directions, and might thus by chance compensate one another 



* Ante, p. 281. t On Knots, Trans. R.SE., 1876-7. 



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