256 REV. T. P. KIRKMAN ON THE K-PAETITIONS OE THE R-GON AND R-ACE. 
The number of angles of the nucleus from which, on one side of a clear diagonal axis. 
e scores can be drawn, is ^{n — 2). Wherefore, by the reasoning of LT., 
RfO-, k). 
LX. When the scored axis is a diagonal one out of an even number of axes, the con- 
structions Rf(r, k)' of Theorem E, are enumerated by a subtraction and division like 
those of LVII. Let e first be even. 
n—2 
The entire number of e-plets eligible from the 
,,_Q\ e|-l 
numerals on one side of this 
-y ) xfe+1 ; and that of g-plets eligible alike on opposite sides 
r+1 , when ?2=4«2-f 2, and • 
' ^ . I 1 1 
^11 . /«— 4\Tr^ |e , T { ^ , N 
2 + 1 •4«-2— 4 ) ■;2 + ^ ■+J-2e • («•) 
of the centre of the axis is 
when n= 4:771: wherefore 
f/n— 2\4-i / — TT"* /n — 2\ 
i[{—) -{—) 
is the number out of Rf(r, k)”, when e is even, that can be made by e-scoiing a clear 
diagonal axis of a 2My reversible (1+^ — e)-partitioned r-gon. 
But when e is odd, a diameter of the r-gon must be one of the e scores, if all is 
symmetrical about the centre, which requires r~4m. Then Avhen w = 4?7i+2, every odd 
e scores that can be drawn gives an unsymmetrical configuration, so that there are thus 
.f^l ‘j, out of Rf(r, >?:)", (b.) 
obtained by so scoring a clear diagonal axis of a 27i-ly reversible. And when 7 i = 4m. 
Ave have merely to subtract from this entire number of Avays as before, the number 
of symmetrical ways, which, after draAving a diameter, is that of draAving ^(6?— 1) scores 
on one side of the centre, and then to divide by tAvo, as before, thus : — 
Then uniting the expressions (a.), (b.), (c.), we find 
•fe + 1 
n—2\ 2 
1 +^ 
iJ +1 
-2._, 
e-1 
n— 4 
2 
I 
r 
for the exact number out of Ef(r, k)" that can be made from any one 2/i-ly reA'ersible 
(1+7:— e)-partitioned r-gon, by drawing e scores upon a clear diagonal axis. 
When the axes of the r-gon are all clear and diagonal, AA^e obtain tAvice this number, 
for we can score an axis carrying either of tAvo configmutions ; but we obtain it once 
