AND RETICULATIONS OF THE R-GON. 235 
quired formule are to be found for the (4+ m)-partitions, 
which have m marginal faces and £+m-—1 diagonals. I 
have however a method of enumerating these partitions 
for all values of m, which is far better for the purpose 
of computation than the general expressions in terms of 
r, k and m would be, if they could be assigned. It is 
evident, from what is here done in the case of m=2, that 
the sought formule will be of very great complexity, un- 
less it can be shown that the series 3’, which have been 
above investigated have some simple involution in which 
# does not appear. When this involution is discovered, it 
will be worth the while to continue the investigation for 
higher values of m. Meanwhile, if the 4-partitions of 
the r-gon are required, without classification as to their 
marginal faces, 1 know of no method of enumeration 
more simple than that which I have given in the memoir 
referred to in the first article, in the Phi/osophical Tran- 
sactions. 
17. The problem of £-partitions of the area of the r-gon, 
viewed in its completeness, is the problem of reticulations, 
in which the partitioning lines may or may not meet in 
certain points within the 7-gon. These points may be 
termed the nodes of the reticulation. The solution of the 
problem of the j-nodal k-reticulations of the r-gon pre- 
sents difficulties not less formidable than those which have 
thus far bid defiance to our analysis in the theory of the 
polyedra. When j=0, we have the question of simple 
partition by 4-1 diagonals; when j>0, there are 7 points 
within the 7-gon, in each of which three or more of the 
partitioning lines meet — some of these lines being diago- 
nals, some passing through a node and an angle of the 
r-gon, and others through two nodes. The & reticulations 
are & smaller polygons, all having only angles of less than 
180°, into which the 7-gon is divided. I have succeeded 
in the discovery of a general inductive method of enumera- 
VOL. XV. II 
