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XII. On the 'K-partitions of the ^-gon and 'R-ace. 
By the Rev. Thojvias P. Kirkmajst, A.M., F.R.S., Rector of Croft with Southworth. 
Communicated by Arthur Cayley, Esq., F.R.S. 
Eeceived November 13, — Eead December 11, 1856. 
I. By the ^.-partitions of an x-gon, I mean the number of ways in which it can be divided 
by ^ — 1 diagonals, of which none crosses another; two ways being different only when 
no cyclical permutation or reversion of the numbers 1 2 3 . . r at the angles can make 
them alike ; and by the 'k-partitions of an x-ace (a pencil of r rays in space or a plane), I 
mean the number of ways in which it can be divided into k smaller pencils, by the intro- 
duction of ^ — 1 connecting lines, of which none enclose a space ; two ways being different 
only when by no cychcal permutation or reversion of the numbers 1 2 3 . . r in the angu- 
lar spaces of the r-ace they can be made identical. The polygon here considered is the 
section of a pyramid, and its discussion includes that of the polyace. 
The enumeration of the partitions of the polygon and polyace is indispensable in the 
theory of the polyedra. In a memoir “ On the ^-edra which have an (^— l)-gonal base, 
and all their Summits Triech’al,” in the Transactions of the Royal Society, 1856, 
page 399, I have investigated the (r — 2)-partitions of the r-ace, or the r-gon; for the 
number of or-edra there determined is exactly that of these (r — 2)-partitions. What 
follows may be considered as a completion of the investigation in that memoir begun ; 
yet not properly a continuation, inasmuch as the results there obtained are here deduced 
by a different and more general method. 
II. A partition of an r-gon is reven'sible or irreversible : reversible, when it is symme- 
trical about a diameter or bisector of the figure, so that the configuration is unaltered by 
a semirevolution about that line, which is called an axis of reversion, of which axes there 
may be one or many; and irreversible, when it is reversible about no axis. An irre- 
versible is about no axis its own reflexion. 
An axis of reversion is always a bisector of the r-gon, and is agonal, monogonal, or dia- 
gonal, according as it passes through no angle, one angle only, or two angles of the r-gon; 
and the polygon is said to be about that axis, agonally, monogonally, ox diagonally 
reversible. 
A diagonal axis may be drawn or undrawn; a monogonal or agonal axis is always 
undrawn. 
III. A partition is said to be m-ly reversible when it has m axes of reversion. The 
simple 2?i-gon (^=1) is 2n-\j reversible, having n agonal and n diagonal axes; and its 
sides may be so loaded with polygons, that this number of axes shall be either retained 
or diminished. The simple (27i-j-l)-gon has 2n-\-\ monogonal axes, on which an r-gon 
MDCCCLVII. 2 G 
