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



of results becomes unmanageably vast long before n the number of crossings is 

 out of its teens. 



29. The final operation, after construction of all knots of n crossings with- 

 out concurrences, is to take every subsolid and unsolid E, in our lists which has 

 n — c crossings and no concurrence, and to add to it in every possible different 

 way c flaps making with one or more on R concurrences of every possible 

 number of two or more flaps, thereby adding c crossings, and completing the 

 number n. This last operation soon becomes impracticable from the number 

 of results. 



30. Nothing can be here added that will give so much insight into our 

 subject as the actual construction of knots, to which we now proceed, first to 

 that of subsolids, and next to that of unsolids of the number n in hand of 

 crossings. 



Two Fundamental Subsolid Knots. — The only subsolids that cannot be 

 reduced by deletion of a leading flap (art. 19), are those of three and of four 

 crossings. These, 3 Aand 4 A( vide Plate XL.), are irreducible and fundamental. 

 3 A is a 3-zoned monarchaxine, whose principal poles are triangles not plane, 

 which have three common summits and no common edge. 



The unsolid 4 B is formed on 3 A by art. 29, and has a symmetry of like 

 description. The secondary 2-zoned poles on either are alternately flaps and 

 crossings, being heteroid poles in 3 A and janal in 4 B. 



31. Subsolid and Unsolids of Five Crossings. — The subsolid must be built on 

 3 A. The only points that can be here joined by a flap, are either on one flap of 

 3 A or on two. We cannot obtain a subsolid by joining the former pair, because 

 the constructed knot would be an unsolid having a concurrence of two (art 13). 

 We join the latter pair, and it matters not whether we draw our flap in the 

 upper or in the lower of the two triangles whose summits are the same three 

 crossings, and whose edges are different halves of the three flaps of 8 A. 

 Drawing the flap 54, the two flaps of 3 A connected by it become triangles, and 

 S A is constructed, a 2-zoned monarchaxine heteroid, whose zoned poles are a 

 tessarace and an opposite tetragon. This is the only subsolid of five crossings. 



The unsolid 5 B is by (29) formed on 4 A, and 5 C is made on 3 A. 



32. Knots of Six Crossings. — The subsolids 6 A, &c, must be formed on t A 



and 4 B. This 4 A has a janal 2-zoncd axis through the 

 centres of the flaps, and two like 2-ple janal zoneless axes 

 through two pairs of opposite mid-edges. It has only one 

 mesh, the monozone triangle 342, and the only pair of points 

 that can be joined are 5a and 56. 



Drawing 5a, or rather conceiving it drawn, we write 

 to determine the leading flap, 



(5a)=43,43; (12)=43,43; (5a)>(12)? 



