220 REV. T. P. KIRKMAN ON THE PARTITIONS 
XV.— On the Partitions and Reticulations of the r-gon. 
By THE Rev. Tuo. P. Kirkman, A.M., F.R.S., 
Rector of Croft-with-Southworth, and Honorary Member 
of the Literary and Philosophical 
Society of Manchester. 
Read March 22nd, 1859. 
1. AttHoven I have given the formule of an inductive 
method (Philosophical Transactions, 1857, “On the parti- 
tions of the r-gon and the r-ace”’), whereby the partitions 
of the 7-gon made by any & diagonals, none crossing 
another, may be enumerated, much remains to be done 
before they are expressed directly in terms of 7 and &. 
The case of kK=r—3, that of the triangular partitions, has 
been discussed in a previous part of the present volume 
(p. 43), in which this direct expression has been investi- 
gated. The most convenient classification seems to be 
that according to the number of marginal faces; it is, at 
least, essential in the theory of the polyedra. 
In this Paper will be given the required expressions of 
the partitions of the 7-gon by *+1 diagonals, in which 
there are two marginal faces. 
2. Such a partition, F, will always have 2+2 angles 
unoccupied by a diagonal. Let w of these —all except 
one in each marginal face -- vanish, that is, become angles 
of 180°. F has now become IF’, a (k+ 2)-partitioned 7’-gon 
(r’=r-—«), having two marginal triangles, and no angle, 
besides their two vertices, unoccupied by a diagonal. No 
face, either of F or of F’, has more than two diagonals in 
