252 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-GOX AXD E-ACE. 
If the drawn diagonal axis be erased, the figure still remains reversible about the same 
axis, as in the preceding article, and we may have restored the symmetry about some 
^i-gonal nucleus of which that erased line is a clear diagonal axis. Every /-gon of the 
number 
Ef*(r, k—\\ 
will give us two of for we can draw a diagonal axis bisecting either of its con- 
figurations ; and every r-gon of the number 
which have all a clear diagonal axis at right angles to a diagonal axis (VI. VII. Cor. 1). 
will give us one of E^*(r, by drawing any diagonal axis of reversion, which can be 
draAvn without meeting diagonals ; i. e. 
comprehending all the values of A>0 and in our register of reversibles. 
Hence follows (XLIV.), 
E^^'(8, 6),==E,f(8, 4), =1, 
E^^'(8, 3),=:E|f(8, 2)e =1, 
E^‘'‘(8, l),=Er''*(8, 0)3=1. 
LIV. Problem q. To find E^‘(r, k) 2 , the number of {l-\-\i)-partitioned singly reversible 
r-gons having a drawn diagonal axis. 
If this axis be erased, the figure F remaina reversible about the same diagonal axis, 
and the erasure may have restored the symmetry about some ?«-gonal nucleus of which 
that erased line is a clear diagonal axis, one of an odd number of axes of reversion in F ; 
for if this number were even, there would then be an axis (VI.) perpendicular to the 
erased one, about which the symmetry would not be disturbed by the erasm'e, so that 
the figure before the erasure must have been at least doubly reversible, contrary to 
hypothesis. 
Now every one of the number k—l)^ will give one (Theorems B, G), and 
only one, of E*(r, k)^., by drawing of any of its axes of reversion, because all these carry 
the same configuration. Consequently 
comprehending all values of 7^ including A =0, and all of?^, in our register of ^'-partitions. 
Hence we obtain (XLVI-.), 
5)2=Ef(8, 4), =2, 
3)2=Ef(8, 2)e+Ef(8, 2),=2. 
LV. Problem r. To E““(r, k), the number of {l-{-\iypartitioned r-gons reversible 
only about a single scored mo7iogoQial axis. 
The number of ways in which e diagonals can be drawn at right angles across an axis 
