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



as in the unifilars appears to lie mainly the interest of these knots. The 18 

 plurifils (if ever they deserve a name) can easily be drawn by the student with 

 the aid of the above list ; and they must be figured by him who constructs the 

 10-fold knots, for on many of them flaps can be drawn to make subsolids of ten 



crossings. 



We have found of eight crossings, 



1 solid knot, unifilar, 

 22 subsolids, 8 A . . . 8 W, 

 17 unsolids without concurrences 

 36 unsolids with concurrences ; 



in all 76 8-fold knots, of which 35 are unifilar. 



47. Consfruction of Knots of Nine Crossings. — The subsolids 9 A, &c, are to 



be formed by drawing leading flaps on 7 A, &c. 



The only lines that differ on 7 A here figured are, 



fb and dd, each =44; 



df, dd, db, dg, ge, ee, eb, each =34 : 



2376, 347, 123, and 134 are the only faces. The d's 

 are alike, all on the same asymmetric edge 34, which 



has two different sides. 



(fb) = 44, (fd) = 43, and (dd) = 43, have no competitors ; for this (dd) fixes 



(67) (art. 23), 



(dd)=A4, 44>(67)=44, 44 ? 





All the lines =34, except fd and dd, are led by (67). We have four flaps to 

 draw, expecting symmetry with (dd) = 44. 



By (cW)=44 we get 9 A ; by (W)=43 we get 9 D ; 



(/&)=44 „ 9 B; „ (fd) , 9 C. 



The symmetry and circles of these four knots are read on their figures. 



48. We consider next 7 B, here drawn. This has the faces 1473, 6745, and 



541, with one zoned polar and one epizonal flap. The 

 different lines that can be drawn are, 



ah, aa,fb, bb, and cb, each=53, 

 ab,fc, and bb, each =44, 

 aa and ae, each =43. 



In bib (bb) =53, 55>(23) or (54) each 53 54 ; (67) is fixed (art. 23) 



(aa) = 53 > (23, 67 or 54) each =44; 



(fc)=U >(23)=43; 

 In 6766 (bb) =44, 55>(67)=44, 55 ? (45)=(23)=43. 



