306 REV. T. P. KIRKMAN ON THE ENUMERATION, DESCRIPTION, AND 



63. We have next to add two concurrences of flaps to all of 7 A, &c, on 

 which is no concurrence — 



7 A gives Q Gn ;, 



7 B „ Gp , 9 Gg , 9 Gr , 9 Gs ; 



rj\j „ $*-*£ > ^yxU 5 



7 D „ 9 Gv , Q Gw ; 



7 E „ 9 Gx , 9 Gy , 9 Gz , 9 Ha , 9 H& , 9 Hc ; 



7 F „ 9 Hd , 9 He , 9 H/ , 9 H<7 ; 



7 G „ 9 Hh , 9 Hi , 9 H/ ; 



7 H „ 9 Wc , 9 H7 , 9 Hm ; 



7 I „ 9 H» , 9 Hj? , 9 Uq , 9 H> ; 



7 J „ 9 Hs , B.t , 9 Hm , 9 Hv . 



The number of results in any of the above cases of this article is that of the 

 different flaps which can be made a concurrence of three plus the number of 

 different pairs of flaps that can be made each a concurrence of two. 



64. We have next to place three concurrences of flaps on 6 A, &c, four on 

 5 A, five on 4 A, and six on 3 A — 



C A gives 9 Hw , qH.x ; 







6 B „ 9 H?/ , 9 H2 , Q la 



9 16 , qIc 



ld; 



0^ » 9* e > oV ' 9^9 > 







G D „ 9 I7i , 9 K ; 







6^ ,, a!/ s sJ& • 







5 A gives 9 I/ , 9 Im , Q ln ; 

 4 A » $P > 'M > o Ir ; 

 3 " 9 ■ 



Finally, there is one solid knot, J.t. 



The number of 9-fold knots that have concurrences is 128, of which we 

 have figured only the 70 of them which are unifilars. The rest will have to be 

 drawn if the census of unifilars is carried to higher values. This can easily be 

 done. 



We have found 244 knots of nine crossings, viz. : — 



1 solid knot, 

 63 subsolids, 



52 unsolids without concurrences, 

 128 unsolids with concurrences. 



Of these 244— 



30 + 25 + 70 + 1 = 126 are unifilar. 



