226 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OF THE E-GON AND E-ACE. 
functions, it is a continuous function, and has a permanent form : wherefore it is 
sufficiently evident that 
fp7~2\l 
D(r,r-3)=^. 
We know that D(r, A:)=0, if ^>0, for every value of r from r=3 to r=k—‘l ; for no 
(^— 2)-gon has k diagonals none crossing another. Hence (r— A’— 2)*'* is a factor of 
D(r, k). Therefore 
D(r, ‘(AV"+B'r^-‘+ . . +LV+M'), 
D(y?:+3, ^)=1.2../i:(A'(y?:+3)^+BPH-3)^->+--+L'(^+3)+0), 
=A'(*+3)‘+B'(^+8)*-+ .. +L'(^+3), 
(^ + 3 + l)(^ + 3 + ^^ + 3 + i^ _l) 
[ A + 2 (a + 1 
Hence 
A'= 
L'= 
1.2.3.(^-l) . 
^k + 2\k + \ 
consequently 
>• -V-^ X (k^\k+i .U.t.u. 
XXHI. Theoeem U. Every m-ly irreversible ^-partition of an x-gon occurs 2r : m times 
among the 'k-divisions, and every m-ly reversible k-partition of it is found r : m times among 
the k-divisions. 
For any ?}i-ly irreversible ^-partition has a different configuration about r : m successive 
angles (HI.), and is nowhere its own reflexion ; so that r : m more configurations the 
reflexions of the former are read on the reversed face of the r-gon. And as all these 
configurations are found among the y^-divisions separately enumerated about the same 
angle, this partition is counted 2r : ni times among them. 
And any m-ly reversible ^-partition of the r-gon has r : ni different configurations in 
the interval between the termination of any axis A and its repetition, tliat is, about 
r : m angles if A be an agonal axis, and about r : m sides if it be diagonal. The central 
configuration of these r : m is about the axis alternate with A ; the others form pairs of 
configurations reflecting each other. All these are constructed and counted separately 
among the ^"-divisions about the same point of the r-gon ; i. e. this partition is coimted 
r : m times among the ^-divisions. Q. E. D. 
Let k) denote the entire number of 7?i-ly reversible (Ifi-^’)-partitions of the r-gon 
about all nuclei, and all kinds of axes, and let I”*(r, k) stand for tlie complete number 
of m-ly irreversible (l+^)-partitions. 
The theorem U may be thus expressed : 
D(r, >i0}, or, (»i>0). 
