AND RETICULATIONS OF THE R-GON. 225 
ra 
diagonals. This axis, if it lies, of course an undrawn 
diameter, through two mid-edges of a central face, is an 
agonal axis; if it is drawn or drawable through two angles 
of G, it is a diagonal axis; if it lies, of course undrawn, 
through one angle and an opposite mid-edge, it is a mono- 
gonal axis. 
8. Let the 7-gon G have an agonal axis of reversion. 
By the erasure, as before, of w angles not occupied by 
a diagonal, G becomes the 7’-gon G’, (7) =r-—z), having 
the same axis, and no unoccupied angle except the two 
vertices of the marginal triangles; and G’ becomes the 
(2k+4)-gon G”, if we pare its margin, having k=2i+1 
quadrilaterals and two triangles. Of these k, let all the 
consecutive edges vanish except y, and in these y quadri- 
laterals let all the opposite edges vanish except z, whereby 
the & quadrilaterals are reduced to z. This can be done in 
wi gels (4(k- 1D) ee ye 
en LaLa wi ie 
so that the reduction of G” shall be effected symmetrically 
on both sides of the agonal axis. One of our results will 
be G’, if the spared edges 
ways, 
wherefore the number of figures obtained from G” is 
5 (4(k-1))y"'7 pret ‘—k—5) | —1 
y yyii "ker SS (y > 9). 
This series is easily proved, by the reasoning of Art, 4, 
‘a Ly iy 
to be 
Gieatyyerasnin 
[i@r—k—) | 1 
S Qk—-w'42 
which is the number of r’-gons G’, to one of which we 
reduced G. This one becomes G by the addition of 4x 
points in 47’ positions, viz., about the vertex of a marginal 
triangle, on two opposite half-edges of the central quadri- 
lateral, and on }(r’—6) other edges of G’ all on the same 
