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



himself, without debate about the terms employed, as to the truth or error of 

 my enumerations. 



11. All that we need to add here is on the symmetry of flaps, which are 

 2-gons, correctly drawn as two curved lines through the same two points. 



A flap has the symmetry of its undrawn diagonal d through its two cross- 

 ings, and may be conceived as standing symmetrically about d, in either of two 

 planes at right angles to each other, which contain d. This d may be asym- 

 metric, or epizonal, or zonal, or zoned polar, or zoneless polar ; and the flap is 

 accordingly asymmetric, or epizonal, &c. The two edges of a flap are unlike 

 only when it is asymmetric or epizonal. 



In a zonal flap, a single zonal trace passes through the two crossings ; in an 

 epizonal flap, a single zone passes between the crossings. In a zoned polar 

 flap, two zonal traces intersect in the centre, the termination of the 2-zoned 

 axis. 



In the centre of a zoneless polar flap, an axis of 2-ple repetition termin- 

 ates. 



3 A and 4 A have each one zoned polar flap. 



5 A has one epizonal flap. 



G F has one epizonal and one asymmetric flap. 



6 G has two different polar, and one zonal flap. 



6 H has one zonal flap and no other. 



8 S and 8 Q have each one zoneless polar and one asymmetric flap ; so has 8 A/. 



8 AZ; has one asymmetric flap and no more. 



8 Awi has one zoned polar flap and none other — art. 45. 



& Bn has one epizonal, two zonal, and one asymmetric flap. 



8 Bq has one zoned polar and one asymmetric flap. 



8 Bw has three zonal flaps only. 



s Bv and 8 Bw have each one epizonal and two asymmetric flaps. 



8 B?/ has one zoned polar and one zonal flap. 



12. The construction of polyedra and of other reticulations of n summits is 

 best apprehended by studying their reduction by fixed rules to antecedents or 

 bases of n — i summits. We proceed to the reduction of knots. 



Reduction of an Unsolid Knot Qo/n Crossings. 



In this are two processes, — the clearing away of concurrences, and the 

 removal of least marginal subsolids. 



13. Concurrences. — Two or more continuous flaps, each having a crossing 

 common to the next, are a concurrence of two or more, except when two or 

 three flaps are collateral with the same triangular mesh. Three such flaps on 

 a triangle complete the irreducible subsolid 3 A ; and two on a triangle are not 

 counted a concurrence. 



