PROFESSOR TAIT ON KNOTS. 



501 



II. Census of Tenfold Knottiness. 



17. Omitting composites, the number of separate types of 10 fold knottiness 

 is, as shown in Plates LXXX., LXXXL, 123. Of these 48 are unique, while 

 the remaining 74 give 315 distinct forms, 364 individuals in all. The largest 

 number of distinct forms for one type is 12 ; and there are two such groups. 

 One type which furnishes a group of 10, has 4 of them amphicheirals of the 

 first order and first class, the remainder of the second class. 



Each of the figures is drawn in the special deformation in which it is 

 presented by the polyhedral method ; and, for reference, the corresponding 

 designation of the knot in Kirkman's list is appended to it. 



18. Of the 107 partitions of 20, under the limits imposed by the nature of a 

 knot, 52 only are utilised ; the rest belonging to links, composites, &c. These 

 are as below ; each being followed by a distinctive letter, which will presently 

 be employed (for brevity) in place of it. 



For knots with 6 right-handed and 6 left-handed meshes : — 



Erratum. — Pa^e 501, par. 17, line- 2, for 48, read 49. 



Y<±45Z e 



74333 f 



5/9 



'7 

 8 



55532 q 



4433222 e 



55442 r 



4333322 f 



55433 s 



3333332 n 



54443 t 





44:4:14: U 





7652 f 



43322222 d 



7643 g 



33332222 K 



7544 h 





6653 k 





6554 1 





ler : — 





965 s 



332222222 X 



66422 # 



66332 h 



65522 h 



For 4 of one and 8 of the other 



8732 a 



8633 b 



8552 c 



8543 d 



7742 e 



And for 3 of one and 9 of the other :- 



992 p 



983 q 875 t 



974 r 776 u 



19. In Part II. of this series I arranged the types of each degree of knotti- 

 ness in the order in which their respective deformations first appeared in Mr 



VOL. XXXII. PART III. 4 N 



