CONSTRUCTION OF KNOTS OF FEWER THAN TEN CROSSINGS. 



303 



Here 



(fa) gives 9 Ba ; 



(fb) » 9B&; 



(go) gives 9 Bc ; 

 (gc) „ 9 Bd. 



The circles and symmetry are on the figures. 



56. On 7 L, here given, no line can be drawn to spoil the concurrence and 

 the linear section but from a or b, 



(ae) = 54 > (23) = 44 and > (67) = 53 ; (54) is fixed (art. 22). 

 not (ac)=63 >(23) = 64; 

 not (ad) = 54, 33 > (23) = 54, 43; 

 not (fy) = 53,53>(67) = 53, 54; 



not (If) = 44 >(67) = 53. 

 Here 



(ae) gives us 9 Be. 



On 7 M, here seen, the leading flap must be drawn from a. 



{ad) = 63 > (12) = 44, or (34) = 43 ; 



(a£) = 54 >(12 or 34) = 53 ; (56) is fixed (art. 22). 



(ac) = 54, 43 > (56) = 54, 43 ? (34) = 44. 



Here 



{ad) gives 9 B/, [ab) 9 B^ ; and {ac) 9 B&, the latter symmetrical. 



57. On jN, annexed, 



(ad) = 54 > (42, 26, 17) = 54? 

 (26) leads (ab), and (45) leads (ac). 



The only leader 



(ad) gives us 9 Bi, symmetric. 



On 7 P no flap can spoil both concurrence and linear section 

 On 7 Q there can be drawn only one leading flap — 



(ab) =56, giving 9 B/. 

 On 7 R, here given, 



(«c) = 55>(32 or 17) = 55? 

 (ab) is led by (17). 



Here (ac) gives us 9 B/£, symmetric. 

 On 7 S, annexed, 



(ac) = 55>(76 or 34) = 54. 

 (aJ)=64>(66 or 34) = 64? 



Here (ac) gives 9 B/, and (ab) 9 Bm, the latter symmetric. 

 We have constructed by their leading flaps sixty-three subsolids of nine 

 crossings, of which thirty are unifilars, bearing on their figures the number 18. 



