115 
The method by which these results are obtained is 
perfectly general, is in no way tentative, and involves no 
reference to figures. 
Although I have given formulae whereby the /d-partitions 
of the r-gon, {j=o) can be found for all values of r and k, by 
an inductive method, much remains to be accomplished before 
the direct expression of them in terms of r and k is obtained. 
I have already investigated such general expressions for 
the (r — 2)-partitions of the ?*-gon, i.e. the triangular partitions, 
made by drawing r — 3 diagonals, in the last volume of the 
Manchester Memoirs. 
About the simplest case that can occur, when there are 
fewer than r — 3 diagonals, i.e. when the partition is not 
triangular, is that in which the ;--gon is divided into A+2 
triangles and quadrilaterals, the number of marginal faces 
being two only, both triangles, and every angle but two being 
occupied by a diagonal. 
If this figure consists of two marginal triangles, separated 
by k quadrilaterals, all the diagonals being parallels, there is 
only one way of drawing it. 
Jf the figure has non-marginal triangles, that is, if the 
number k-\-2 of the diagonals is greater than \r — 1, it will, as 
in every other case, either be symmetrical or asymmetrical. 
If there is a symmetry of reversion, it will have a diametral 
axis of reversion, either drawn through two angles, or 
drawable through one angle and the middle of a side, or else 
drawable through no angle and the mid-points of two opposite 
sides. In the first of these three cases, the figure is said to 
have a diagonal axis, in the second a vwnogonal axis, and in 
the third, an agonal axis of reversion. A section along this 
axis of reversion divides the figure into halv'es, of which one 
is the reflected image of the other. 
If the symmetry is not one of reversion, it is one of 
repetition ; that is, there is an irreversible sequence of configu- 
ration read twice in the circuit of the r-gon. 
