EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-G-ON AND E-ACE. 228 
for it meets all those scores singly. Wherefore A is the only axis of reversion of N'. 
Q. E. D. 
XIX. Theoeem If a clear aocis K of a singly reversible partitioned i-gon N' he 
scored by diagonals perpendicular to it, the scored figure N becomes sometimes singly, and 
at other times (2m+3)-Zy reversible about the scored axis. 
For the r-gon N' being singly reversible has no axis of reversion perpendicular to A ; 
and no addition of diagonals parallel to the perpendicular diameter can make it an axis of 
reversion ; for that addition cannot alter its intersections with the diagonals of N' ; 
wherefore N is not 2TO-ly reversible (Theorem D and B). If N' should be the N' of 
Theorem Q, and the scores upon A should be those erased in that theorem, N will be 
the (2m-l-3)-ly reversible of that theorem: but if this is not the case in both these con- 
ditions, N will remain, like N' unscored, singly reversible about A, since the scores do 
not disturb the symmetry about that axis. Q. E. D. 
What precedes about singly reversibles with loaded axes is sufficient for our present 
purpose, which is to show that before we can determine the number of singly reversibles, 
with clear and loaded axes, it is necessary that we should know the number of (2m+l)-ly 
reversible e)-partitions of the r-gon which have a clear axis, ^. e. which have clear 
axes ; for here the configurations about all the axes are alike (Theorem C) ; and also 
that of the 2m-ly reversible (1-f-^— e)-partitioned r-gons which have one configuration 
about clear axes, and also of those which have both their configurations about clear axes. 
This matter will be more evident as we proceed. 
XX. Let E*“^(r, kf, E**(r, k)^, E*’"°(r, k)^ denote the whole number of 
(l-f-^j-partitioned r-gons built on the 7i-gonal nucleus (%>2), which have (Theorem C) 
k)n, h agonal and h diagonal, 
^ agonal only, 
^ diagonal only, and 
R* ’"® (r, k)n, h monogonal axes only, of reversion. 
We shall denote those having all their axes clear by c subscript to R; those having 
no clear axes by zero subscript to R; those of the second and third classes which have 
half their axes, bearing one configm’ation, clear, by ^c subscript, and those of the first, 
which have half their axes clear, the agonal or the diagonal ones, by ac or dc subscript 
We wiite 
. ag, 
*(r, k),-- 
- Rf ■ k \ + R“ • 
RA.fl? 
{r, kf- 
II 
{r, y?:)„+R^r" 
(r. 
(r, 
. di 
{r, k\^ 
II 
{r, R^/' 
{r. 
(r, kf, 
.I7M) 
{v, k\- 
_RA.mo 
{r, -^)„+R^“'’ 
(r. 
Jcl- 
In the second and third lines the second subclass is of course nothing when h is odd. 
In the fourth class the number of axes of reversion is always odd. And in all those 
equations we suppose w<t;3. 
