EEV. T. P. KTRKMAN ON THE K-PAETITIONS OF THE E-G-ON AND E-ACE. 219 
which begin and end with a side or angle carrying an axial termination, and those being 
irreversible which begia and end at any other side or angle. 
Ohs. 3. But a singly reversible partition has no sequence repeated in the circuit of the 
i^-gon ; for if it had a repeated sequence reversible as read from no point, the r-gon would 
not be reversible ; and if it had a reversible sequence repeated in the circuit of the r-gon, 
it would not be singly reversible. 
Ohs. 4. A 2m-ly reversible r-gon, if r>2m, has beginning at any angle of the r-gon 
which is not the termination of an axis, an irreversible sequence, simple, if m=l, and 
m-ple, if m>l, occupying half the circuit of the r-gon, and repeated in the other half. 
VI. Theoeem B. When the number of axes of rexersion is odd in any partitioned x-gem., 
none is perpendimlar to another ; and when that number is CTm, every one is perpendicular 
to some other. 
For when the number is odd, there is on each side of any one a an equal number of 
terminations of other axes, all equidistant from a and from each other. And when that 
number is even, there is an odd number there of such terminations. Whence the truth 
of the proposition is evident. 
VII. Theoeem C. When the axes of reversion are odd in any partitioned r-gon, each 
one bisects both axial configurations ; and when they are even in number, each bisects but 
one, read on it alike at either end, and half the axes carry one, and half the other, axial 
configuration. 
This is very evident from what is proved in Theorem A, that the axial configurations 
present themselves alternately upon the axes in order. 
Cor. I. If there be both agonal and diagonal axes, there is an equal number of each 
kind ; and, as this number is even or odd, so is each axis perpendicular to one of its own 
or of the other kind. 
Obs. 6 . A (2m-l-I)-ly reversible partition never has an irreversible sequence occupying 
half the circuit of the r-gon and repeated in the other half; for this would require that 
every axis of reversion should carry the same configuration at both ends, which are points 
in those two sequences. 
VIII. Theoeem D. If a diagonal be perpendicular to an axis of reversion in any \i-ly 
reversible r-gon, it is one of a system of not fewer than h diagonals symmetrically placed 
about the centre. And all diagonals not perpendicular to that axis form pairs making 
each an angle bisected by that axis, or that produced. 
For if A be odd, every axis canies the same perpendicular; and if h be even, at least 
^h axes carry that perpendicular on opposite sides of the centre ; and these axes are equi- 
distant from each other : whence the first part of the theorem is evident. The second 
part follows from the definition of an axis of reversion. 
Cor. The intersection of two produced diagonals equidistant from the centre, in any 
reversibly partitioned r-gon, is upon an axis of reversion, 
IX. Theoeem E. No {2rri-\-l)-gon has an agonal or diagonal, and no {2m)-gon has a 
monogonal, axis of reversion. 
2g2 
