254 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OP THE E-&ON AND E-ACE. 
Let e first be even, in constructing Ro^(r, h)'- The entire number of e’-plets eligible 
from the numerals on one side of the axis is 
and that of the | -plots eligible alike on opposite sides of the centre of the axis is 
e 
( n— 4\ 2 
when %=4m, and, when w— 4m+2, it is 
(t) 
- 1-1 
xl^+i , 
1-1 
X '2+1 5 
wherefore, using the circulators 4„ and 4„_2 and 2^, 
Sef/n— 4\®i“' 1 — tT"' /w— 4 X 2 ' ' [e ' /n — 6 X 2 ' ‘ fe~T \ , . 
2 ) ^ ^ j • 2+1-4«~ \ 4 ) •|2+^-4^-2|- • 
is the number out of R7(r, A:)", when e is even, that can be made from any (l+i?' -—el- 
partitioned r-gon having an w-gonal nucleus, 2A clear axes, and clear agonal ones, by 
scoring any one of these with e parallels at right angles to it. 
But when e is odd, a diameter of the w-gon must be one of the scores if they be sym- 
metrical about the centre, which requires w=4m-l-2. Hence, if w=4?/z, ever}' odd e 
scores that can be drawn gives an unsymmetrical arrangement, so that there are thus 
(i) 
out of Ilo^(r, TcY obtained by so scoring any clear agonal axis of a 2A-ly reversible. 
If w=4to+ 2, and e is odd, we are to subtract from the whole number of ways of 
drawing e scores that of the ways of doing it symmetrically, which, when we have dra-wn 
a diameter for one, is that of the ways of drawing ^{e—1) scores on one side of it. This 
difference, divided by two, is 
e - 1 . 1 
''n— 4\®l"^ 
(c.) 
which is the number of ways of drawing an odd number e of scores across the agonal 
axis of a (4w-l-2)-gonal nucleus. 
Uniting the expressions (a.), (b.), (c.), we find 
-2.-.(4...C-=^) ’ 
6 + 1 
for the correct number out of Bo^(/’, k)” that can be made by drawing e diagonals of the 
