484 REV. T. P. KIRKMAN ON 



by drawing in art. 53, in the base under 7, the flap 35,44. But no unifilar of 

 ten crossings can be made on this 9 Ar' 2 . 



3. As Professor Tait excludes all compound knots, i.e., all that can be cut by 

 a closed curve in two mid-edges only, the name of a fixed flap ought to be given 

 to every flap whose deletion lays bare such a compound, i.e., such a section 

 through two edges only. As the deletion of such a flap is forbidden, so must 

 be the drawing of it ; and it cannot compete for the leadership with a flap 

 drawn or about to be drawn. 



A correction is to be made also in (2) of art. 27, which ought to stand 

 thus : — 



(2) If neither e nor e' be zoned polar, but be (a) one zoneless polar and the 

 other epizonal, or (b) one zoneless polar and the other zonal, or (c) one epizonal 

 and the other zonal, only one resulting configuration is possible : in all other 

 cases, when neither e nor e' is zoned polar, and not both are asymmetric, two 

 and only two configurations can and must be made by the above variation of 

 posture of the charge. 



In my plates in volume xxxii. part ii. a few errors require correction. 

 In PI. XLL, for 9 H ; 8, 10 ; read H ; 4, 4, 10 : in PI. XLII., for ,Bi/, 18 ; read 

 9 By ; 8, 10 : for 9 D6 ; 4, 14 ; read 9 D6, 4, 4, 10 : for 9 Ch, 18 ; read 9 CA ; 4, 14 : 

 for Dk, asym. ; read Dk, Moz : after 9 D/ and after J)m write 18. In PI. XLI1I. 

 under Ql, for asym. write 2zo. Mox. Het. 



Postscript, July 13, 1885. — This day I see for the first time that when the 

 problem is to construct, not all the knots of n crossings, but only the non- 

 compound unifilars, in which the tape passes over and under itself alternately 

 at successive crossings, there is no need to discuss at all marginal dissections, 

 nor marginal charges, nor any use of bifilar bases. This is shown as follows :— 



Let K„ be any non-compound unifilar of n crossings alternately under and 

 over all through the circuit. Going round the circle, plant at every mid-edge 

 between two crossings a dot on the right of the thread. 



Every flap will have two dots, both inside, or both outside, or one inside 

 and the other outside of it. In the last case call the flap odd ; in the others, 

 oven. 



The following theorems are easily proved : — 



Theorem A. — If K„ above defined has an even single flap (of one loop 

 only), it can be reduced to an unifilar, solid or unsolid, of n-\ crossings by 

 shrinking up that flap to a point. 



Theorem B. — If K„ has an odd single flap, it can be reduced to an unifilar, 

 solid or unsolid, of w-2 crossings by effacing the two edges and the two 

 summits of that flap. 



Theorem C. — If K has a double flap, of two loops, the two terminal con- 



