218 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-OON AND E-ACE. 
may be built to have the same or a smaller number. To help our conceptions, -we may 
always suppose our r-gon regularly inscribed in a circle ; but it is evident that the syn- 
typy of two identically-partitioned r-gons in no wise depends on such symmetry^ but may 
remain after any distortion of either r-gon which does not change the angles on any dia- 
gonal. So that, if we wish to build an /-gon on an inscribed r-gon, we need not fear 
exceeding the limits of the circle by our additions, while we may suppose these all con- 
tained within it. 
A partition is said to be m-ly irreversible, when it has an irreversible sequence of con- 
figuration m times repeated in the circuit of the r-gon. This sequence will occupy - 
angles ; and, from whatever angle we begin to read, we shall see a sequence of - sides 
irreversible, such that through the mid-point of it no axis of reversion can be drawn. 
Ohs. 1. Hence a 2m-ly irreversible has an irreversible sequence, simple if ?«.=1, and 
yn-ple. if to> 1, occupying half the circuit of the r-gon; but a (2?/i+l)-ly irreversible 
has no repeated sequence occupying half its circuit. 
TV. Theoeem a. Every reversible {l-\-\.)-partition of an r-gon has two reversible 
sequences of configuration which are bisected by alternate and equidistant axes of rever- 
sion, and has not more than two, whatever be the number of these axes. 
For, first, let there be only one axis of reversion in the r-gon : there must be two 
aspects of configuration observable from opposite ends of that axis, otherwise the figure 
would be reversible about a perpendicular to that axis, i. e. there would be two axes, 
contrary to hypothesis. 
Secondly, let there be more axes of reversion than one ; any axis a bisects an aspect 
A, because the figure is unchanged by a semirevolution about that axis ; and the axis b 
next in order to a along the circumference bisects an aspect B. This B is difierent from 
A ; for if not, the series of configurations read from atob will be that read fi:om b to a, 
and there will be either a vertex or a side centrally placed between a and b, having on 
both sides the same aspect, or an axis of reversion can be drawn between a and b ; but b 
is the next in order to a, which is absurd; therefore B is not A. Now ^ bisecting the 
aspect B must have the axial termination a at the same distance on either side of it, and 
for the same reasons a must have the axial termination b at the same distance on either 
side of it ; so that the terminations of the axes must recur at equal distances in the order 
..ababa... bisecting the. aspects ..ABAB.. And this series of aspects has as many 
terms as there are axial terminations, viz. Am terms, if the number of axes is even, and 
Am-\-2 if it be odd. Wherefore no aspect different from A and B can be bisected by 
any axis, and A and B, different reversible aspects or sequences, ai’e bisected by alternate 
equidistant axes. Q. E. D. 
We may call A and B the two axial configurations. 
V. Obs. 2. A reversible partition of the r-gon having more than one axis of reversion, 
. has both reversible and irreversible sequences repeated in the circuit of the r-gon, which 
occupy an interval equal to that between alternate axes ; those being reversible sequences 
