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



54. On 7 H, here seen, we can draw two flaps only, 



(a&) = 64 (in 566a)>(43) = 63 or (21) = 43. 

 (ah) = 55 (in 5625a) > (43) = 53 or (21) = 43. 



One (ab) gives 9 As, the other gives g At. 



Four flaps can be drawn on 7 I annexed, 



(a&) = 64>(56) = 63 or (13) = 53; 

 (db) = 55 > (57) = 53 or (13) = 53; 



(ac) = 55 > (24 or 56) =53 and >(13) = 54; 

 («Y) = 64>(57 or 24) = 63 and >(13) = 54. 



Here (ab) gives Q Au ; (ac) gives Q Aw ; 



(db) 



» 9 



Av: 



(do) 



» 9 



Ax. 



Professor Tait does not allow 7 H and 7 I to be different knots, giving a reason 

 at p. 158, Trans. R.S.E. 1876-7, which appears to me sufficient wherever it can 

 be verified without twisting the tape, or breaking the law of alternate over and 

 under. It is true that on the knot in space whose projection is 7 H, the three 

 crossings 543 which are found on the thread 67 between 6 and 7, can by 

 slipping of the thread be made to appear on the thread 71 ; so that the order of 

 the crossings shall be changed from 165435437 . . , the thread passing over at 

 15347, to 167543543 . . , the thread passing over at 17534, i.e., making two 

 consecutive overs at 7 and 5. The resultant figure in space, although it would 

 have 7 I for its projection, would, if I am in the right, be no knot. If I had not 

 drawn both 7 H and 7 I, I should have missed some unifilar 9-fold knots, both 

 here and in art. 63. 



55. On 7 J, annexed, can be drawn only two lines to make a subsolid, 



(ab) = 54, 44(67) =54,44? (54 = 12) = 43. 



(ac) = 54, 44> (12 or 54) = 43; (67) is fixed. 



(ab) gives 9 Ay, and (ac) gives 9 Az. 



On 7 K, here seen, can be drawn flaps only from /, g, h, or i, so as to spoil 

 »s both concurrence and linear section, 



(fa) = 54, 43 > (54) = 54, 33; (12) = 44; 



(fb) = U >(54) = 44; (12) -43; 



(#f() = 54,43>(57)>54,43? (12) = 44; 

 (r/c) = 54 > (57) -44, or (12) = 53. 



A flap {/</) can be drawn, but the linear section 74 would remain. For the 



rest, 



(45) leads fc, he, hd ; (75) leads by, ic, id. 



