The K-partitions of the R-gon. 109 



On the k-partitions of the R-gon. By the Rev. Thos. 

 P. Kirkman, M.A., F.R.S. 



(Received, October ijth, 1893.) 



A tape-face in a partitioned R-gon is either a triangle 

 having only one edge, or a quadrilateral having two opposite 

 edges, in the contour of the R-gon. No tape-face carries a 

 marginal triangle. 



The first step in the reduction of the general k-par- 

 titioned R-gon is to drop out all its tape-faces. 



The 14-gon, Fig. 9, has four tape-faces. By dropping 

 them out, we make it the 10-gon, Fig. 10, which has no 

 tape-face. 



Fig. 9 has thus lost 4 contour edges, and 3 diagonals, 

 and therefore three faces. 



The last operation in the construction of a definite 

 partitioned R-gon, is the insertion of the tape-faces that it 

 is intended to contain. 



Our definitions and reasonings, until we come to handle 

 the tape, Fig. 8, apply to partitions in which there is no 

 tape-face. 



Definitions : The bases of all marginal triangles of a 

 partition are marginal diagonals. 



All other diagonals are non-marginal. 

 A prime partition is one whose diagonals are all marginal. 

 Figs, t, s, a, b, c, d, e, f, 6, 7, are primes. 



A sub-marginal face has for an edge one, and only one, 

 non-marginal diagonal, and carries z>i marginal triangles. 



A belt is a row of primes only, which have each more 

 than two marginal triangles, and cohere each with the next 

 by the united bases of two marginal triangles, that are 

 H 



