no The Rev. Thos. P. Kirkman on 



hidden, being both creased under. 3Bi is a belt of 5, Fig. 10 

 is a belt of 2, primes. 



We are to conceive two marginal triangles creased under 

 every non-marginal diagonal, in every partition. 



When a belt falls asunder into its component primes, 

 the undercreased marginal triangles are seen in their right 

 position. Compare ABGD1D2 in $$1, with ABabd out 

 of 3Bi, in the figures. 



In JBi, A and D 2 are submarginal, but not in Figs. 1, 2, 



3> 4, 5- 



The belt 3Bi has 41 summits, besides the two terminals 

 crossed, and has 19 faces. 



2. Every face of a partitioned R-gon, which has two 

 and only two non-marginal diagonals dd for edges, maybe 

 and will be here considered, as a loose pane, and may fall 

 out with its complete contour and fringe of marginal 

 triangles, after which the two d d\ whether they have or 

 not a common point, can become one ; so that the number 

 of non-marginal diagonals in the R-gon is diminished by 

 one. 



When a face has for edges 3 + 2 non-marginal diagonals 

 it cannot fall out, so that the 3 + 2 non-marginal diagonals 

 can become one, diminishing thus by one the number of 

 non-marginal diagonals in R. In Fig. H is no face that can 

 drop out. In JB any one of A B Ci Di D 2 may so fall out ; 

 and any two, or all the five, can disappear with their mar- 

 ginal triangles. If all the primes that have for edges only 

 two non-marginal diagonals were so to fall out of Figs. 

 (1, 2, 3, 4, 5), nothing would be left of all the five but exactly 

 our first figure H. 



3. This H is irreducible, because it has no face, having 

 two, and only two, non-marginal diagonals for edges. 



Definition : A k-partitioned R-gon in which is a face 

 that has for edges 3 + 2 non-marginal diagonals, but no 



