AND RETICULAT{LONS OF THE R-GON. 229 
fractions are to be accounted zeros. Here r, & and 2 are 
all even. 
11. If we erase the diagonal axis in H, we obtain an 
r-gon H,, having the same two marginal faces, & diagonals, 
and an undrawn diagonal axis. If R“(r,k)” denote the 
number of such figures H,, we have 
R“(r,k)"=R“(r,k+1), and 
R“(r,k+1)’=R8(r, £42)’, 
which differs from R“(r,4+1)' only in having £+1 for &. 
That is, 
R"(r, b+ 1)" 
= yl OG.) ee ak Eee Qk-Hr—2) +8 
=x? [ira 11 : [hee 11 ; 
Rely 
_Qr-2k-2)) FP], 
1a! ) 
in which 2 has every even value, so that 
ato, et(r-2k-6), w(r-k-5). 
Here & is odd, r and # are both even, and, as before, 
irreducible fractions are to be accounted zeros. 
12. The class I’(r,4+1) of r-gons L is next to be con- 
sidered. These have £+1 diagonals, two marginal faces, 
and a sequence of configuration once and but once re- 
peated in the circuit ;-and they have no axis of reversion, 
so that the upper face is not identical with the lower. 
Snech a figure L has +2 angles not occupied by dia- 
gonals. If x be erased, leaving only those in the marginal 
triangles, the figure becomes an r’-gon (r’=r—x) L’ which 
has still a sequence once repeated in the circuit, and may 
or may not have two axes of reversion; for every one of 
R*(r',k+1) has such a sequence. If now we pare away 
the margin of L’, we ebtain the (24+4)-gon L” having & 
quadrilaterals and two marginal triangles. 
18. When & is odd there is a central quadrilateral in 
L”. Of the 4(4—-1) edges between that and the marginal 
