AND RETICULATIONS OF THE R-GON. 227 
pass through the marginal faces, for this axis has the 
same configuration at both extremities; while each of 
R“(r+1,4+1) will give two figures K, by the vanishing 
of an edge carrying the agonal axis, for this axis has 
different configurations at its two extremities. That is, 
(r+1=2k+4m) 
R”(r,£+1)=R(r4+1, 441) 4+2R%(r4+1,k4+1). 
We have just found that for > =2h+ 4 
(R?+2R”)(r, £+1) 
(ar 2s Yi |—1 (4(& bos 1))i#-*-) ht 
— 2 a ; e , Dk—h(r—2x) +2 
=2, Jr) 11 [30-7—k5) | 1 2 
=R”(r-1,k+1); 
wherefore 
R”™(r, &+1) 
QE= DPI GG) 
=2, pea % [20-2 11 5 
where x+0, e<+tr-2h-8, xpr-—k-4, and fractions irre- 
ducible are counted zeros. That is, & is odd, 7 is odd, and 
# is even. 
10. We are next to enumerate the 7-gons R“(r, £+1), 
which have £+1 diagonals, a diagonal axis, and two mar- 
ginal faces, which axis may be either drawn or undrawn. 
Let H be one of these figures having a drawn diagonal 
axis. It will have v+2 angles not occupied by a dia- 
gonal, by the vanishing of which H becomes H’ an 7’-gon 
(r’=r-—~) having the same axis and two marginal tri- 
angles. By removal of the margin, H’ becomes H”, a 
(24+4)-gon having & quadrilaterals and two marginal 
triangles. Of the 4% edges of these quadrilaterals that 
make one-fourth of the circuit of H’”, excluding the tri- 
angles, let all but y vanish, and in these y quadrilaterals 
let all the opposite edges vanish except z. The same 
changes being made in the other half of H”, so as to 
maintain the symmetry about the axis, H’ will be one of 
the resulting figures, if the spared edges 
VOL. Xv. HH 
