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



tive, and involves no error, here to consider them as cases of the linear sections 

 ffcandtf. 



35. We have constructed all the knots of six crossings that are without a 

 concurrence, viz., 6 A, C B, 6 C, 6 D, 6 E. Those having concurrences are obtained, 

 G F on 6 A, 6 G- and G H on 4 A, and J on 3 A. These nine, 6 A . . . 6 I, along with the 

 solid knot 6 J, are the ten jDOssible knots of six crossings. Four of them, as 

 Tait has found and drawn them, are unifilar, viz., 6 A, 6 E, 6 F, 6 G; and this is 

 read on the figures. The number 12 on each shows that there are 12 steps in 

 the circle of the knot, which passes twice through every crossing, once over 

 and once under. 6 B, 6 D, and G H are bifilars ; 6 C and 6 J are trifilars. 



36. Knots of Seven Crossings. — The subsolids 7 A, &c, must be built on 5 A, 

 &e. The only lines that can be drawn on 5 A here 

 given are ff and aa, each 44 ; and af, ae', ee' he, oh, 

 each 34. 



By ff, which has no rival, we get 7 A ; whose 2-zoned 

 poles are the flap and the tessarace 3333, 



(«a)=44>(12), or (34) = 43; 

 (a/)=53,43>(12)=53,43? 

 (6A)=43, 43 ><12) =43,43? 



For the; rest, ae' , he, and ee' , 



(43)=53>(«e')=43; 

 (43)=44>(5e)=43 and >(ee')=43. 



We 'have to draw, besides ff, the flaps (aa) (af) and (bh), expecting symmetry 

 with the two last, which we soon find. 



By (aa) we get 7 B , 



whose 2-zoned poles are this flap and a tessarace. 



By (af) comes 7 C, monozone ; 



By (bh) 



J), 



whose zoneless 2-ple poles are a 4-gon and a 4-ace. Thus there are four sub- 

 solids, 7 A, 7 B, 7 C, 7 D, reducible by the leading flap to S A. 



37. On 5 B, annexed, as we cannot allow a concurrence, we can draw only 

 (af) and (&/), 



(«/) = 53>(12)=43, and (45) is fixed (art. 22). 

 (Z>/)=44,43>(45)=44,43 ? ((12)= 34), 



for (45) is not fixed when (If) is drawn. 



We have to draw (af) and (hf) looking for symmetry 



in the latter. 



