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XXVII.— On Knots. Part III. By Professor Tait. (Plates LXXIX., 



LXXX., and LXXXI.) 



(Chapter I. read June 1st, Chapter II. July 20th, 1885. One change, small but important, was made during printing. 



It is described at the end of the paper. ) 



The following additional remarks are the outcome of my study of the 

 polyhedral data for tenfold knottiness, which I received from Mr Kirkman 

 on the 26th of last January. My main object was, as in the first chapter of 

 Part II., to determine the number of different types ; as well as the number 

 of essentially different forms which each type can assume, as distinguished 

 from mere deformations due to the mode of projection. 



This study has been a somewhat protracted one, in consequence (1) of the 

 great number of tenfold knots ; (2) of the very considerable number of dis- 

 tortions of several of the types, many of which are essentially distinct while 

 others present themselves in pairs differing by mere reversion ; and especially 

 (3) of the fact that the polyhedral method often presents some of the distinct 

 forms of one and the same type projected from essentially different points of 

 view (of which, in the present case, there are sometimes twelve in all). Reason 

 (3) depends on the fact that Kirkman's method occasionally builds up various 

 forms of one type on different bases of a lower order, and it really involves 

 additional labour only ; but great care is requisite to avoid confusion as regards 

 (2), and in consequence I may not have fully reduced the final number of 

 distinct types. [At the end of this paper I shall give a simple illustration of 

 the nature of this special difficulty.] 



The fact that I was dealing with knottiness of an even order induced me to 

 commence the testing of the materials at my command by picking out the 

 Amphicheirals. This led to some new considerations of a very singular nature, 

 which are treated in the first of the following chapters. The second deals 

 with the tenfolds as a whole. 



I. Various Orders and Classes of Amphicheirals. 



1. As one form of check on Kirkman's results, I sought for an independent 

 method of forming all the amphicheirals of a given order. But, as will be 

 seen below, we must be careful in this matter, which is not so simple as I first 

 thought. I therefore commence by recalling the original definition of an 

 amphicheiral. 



VOL. XXXII. PART III. 4 M 



