220 EEV. T. P. KTREMAX ON THE K-PAETITIONS OE THE E-OON AND E-ACE. 
For a monogonal axis must have an equal number of vertices of the r-gon on each 
side of it, besides the vertex through which it passes ; hence r is odd. And a diagonal 
axis must have an equal number of vertices on either side, besides the two through which 
it passes. And an agonal axis, which passes through no angles, must have an equal 
number of vertices on either side. Hence, in these two latter cases, r is even. 
X. Theoeem F. If one axis of f eversion is monogonal in a partitioned r-gon, all its axes 
of reversion are monogonal and odd in number. 
For r is odd, and the r-gon cannot have either a diagonal or an agonal axis ; and 
as each axis bisects two aspects, A and B, A must be opposite to B in the circle 
..ABABAB.. of axial configurations ; i. e. the number of its terms is 2(2?7Z+1), where- 
fore the axes are odd in number. 
XI. Theoeem G. If there he a drawn axis of reversion., a, in a {l-{-'kypa-ii:itioned 
x-gon., there cannot he more than one other axis. If there he another., b, it is undrawn, and 
perpendicular to the former, a, and is either agonal or diagonal, as r=4m-f-2, or r=4m. 
For, if there be a second axis, b, it cannot meet the drawn one, and must be undrawn. 
And all the k diagonals are symmetrically placed about or upon this h ; therefore a, meet- 
ing it and bisecting it in the centre of the r-gon, meets it at right angles ; other^fise 
(Theorem D) two diagonals would meet h in the centre, which is impossible. And no 
line besides h can so meet a ; wherefore a and h are the only axes. As a is not a mono- 
gonal axis, neither is h (Theorem F). If r=4m, a has on either side an even number 
of sides of the r-gon, and b, bisecting that system, is diagonal; if r=4??i-{-2, a has on 
either side an odd number of sides of the r-gon bisected by h, which is therefore an 
agonal axis. Q. E. D. 
XII. Theoeem H. If there he more than one undrawn axis of reversion in a partitioned 
x-gon, the x-gon is built regularly on a polygonal nucleus (Q), which is reversihle about all 
the axes of reversion of the x-gon, and has no drawn diagonal. 
For consider the symmetry of the r-gon about any one of its axes, a, which are all 
undrawn (Theorem G). We see, on each side of a, f marginal faces, hmited each by one 
diagonal d and certain sides of the r-gon, the 2f diagonals d forming pahs making angles 
bisected by a. Let these 2/ faces be erased : the 2/’ lines d are now sides of an /-gon 
(r'<r), which is still reversible about a, for the symmetry about a is not distmhed by 
the erasures. This r'-gon has also 2f marginal faces which can be erased, leading an 
r"-gon (r" <r') still reversible about a ; and thus by erasure of all the pahs of faces about 
a, the r-gon will be finally reduced to a polygon P, hawng no diagonals but what are 
bisected at right angles by a (Theorem D). Let Q be that portion of P which contains 
the centre of the r-gon, and Qi, Q^, &c. the remaining portions of P limited by perpen- 
diculars to a. Then Q has evidently no diagonals. 
Next consider the symmetry of the r-gon about any other axis h. We can reduce it 
by erasure of pahs of faces about 5 to a polygon P', consisting of a portion Q' about the 
centre and having no diagonals, and of portions limited by diagonals perpendicular to h. 
Q and Q' are polygons about the centre having no diagonals ; they are therefore one 
