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



This S in construction of Q' was counted as a least marginal charge of k—2 

 crossings. Two crossings are always lost when in construction a subsolid 

 having k crossings is imposed at a section ffc. 



In reduction at this section ffc, this S is a least marginal subsolid of k—2 

 crossings along with others of k — 2 removed by any section ff, fe, &c. 



17. Our unreduced Q' of n—i crossings may have in it every kind of least 

 marginal section, art. 15. Such section, not ffc, always cuts two edges, making- 

 four ends. In every case,ff,fe, ef, or ee, those pairs of ends are conceived to 

 be united on either hand into one edge, by which is restored the half-flap or 

 the edge cut away in each portion when united by construction, in order that 

 every summit should be a tessarace in the completed unsolid. 



The imposition of a least marginal charge by ffc costs four edges of two 

 flaps ; by ff, fe, ef or ee it costs only two edges, one on each of the united 

 knots. 



No crossing is lost when a charge is imposed by a section ff,fe, ef, or ee. 

 All this will be found very clear and easy when we come to constructions, and 

 examples will abound. 



18. In. this reduction of Q' all least marginal subsolids, say of k crossings, 

 are to be removed without regard to the number of their meshes, which may 

 differ while each has added k crossings. And care is to be taken that none is 

 cut away which has been loaded with another either on flap or edge, and thus 

 made non-marginal. 



When our Q' is thus reduced, it has become Qx an unsolid of n— j crossings 



U>i)- m 



This Q x will in general have one or more concurrences due to the flaps sub- 

 stituted for subsolids removed. All these are to be cleared away (art. 13), 

 whereby Qx becomes Qi", an unsolid without a concurrence ; and Qi is to be 

 treated as we treated Q' in art. 14. 



We shall finally arrive by these reductions either at an unsolid of two 

 portions, neither of which is least, which is to be reduced by a final section to 

 two subsolids each of c crossings ; or at a ring of flaps which is reducible to the 

 fundamental 3 A ; or to a nucleus subsolid or solid knot. We have now to set 

 about the reduction of subsolids. 



19. Reduction of a Subsolid of n Crossings by its Leading Flap. — The rule 

 is — Remove both edges of a leading flap, or of a leading flap when there are 

 co-leaders. By this removal, the two meshes covertical with the deleted flaps 

 lose each a crossing ; and if one or both coverticals are triangles, that one or 

 both become flaps. The result obtained is a subsolid of n — 2 crossings. 



20. Every flap can be written AB,CD, where A and B (A>B) are col- 

 laterals, and C and D (C>D) are coverticals, of the flap. 



We compare first the collaterals of the flaps whose leader is to be found. 



