PARTITIONS OF THE X-ACE AND THE X-GON. 47 
reversible has six such positions in three axial planes 
making equal angles with each other. All this requires 
the supposition, which we are at liberty to make, that the 
base of the polyedron is a polygon regularly inscribed in 
a circle, on which the whole figure is constructed as sym- 
metrically as possible. 
The three following are irreversibles, single, double and 
triple, standing on 7-gonal, 6-gonal and 9-gonal bases : 
| 54363538, 435435, 643643643, 
of which the third e.g. shows a hexagon, a 4-lateral and a 
triangle thrice read in that order about the base. 
The three following are reversibles, single, double and 
triple, on 5-gonal, 8-gonal and 6-gonal bases : 
53443, 53635363, 535353. 
6. A sequence four times read can never appear if the 
summits are al] triedral. For in such a sequence the tri- 
angles may be supposed to become infinitely small, and in 
the result of their evanescence the triangles again might 
be supposed to disappear, and so on till a figure appeared 
incapable of further reduction by the evanescence of tri- 
angles. But this figure would still have a four-fold se- 
quence, the original periods of configuration having been 
treated alike: that is, the figure would be a pyramid in a 
quadrilateral base, having a 4-ace, contrary to hypothesis. 
The effect of the evanescence of a triangle is to reduce 
by one side the rank of the face on either hand; thus, by 
the vanishing of the triangles, 
643643643 becomes 535353, 
and 53635363 becomes 3434. 
7. PROBLEM a. 
Let P be singly irreversible (2a + 1)-edron, having a 2a- 
gonal base and ~# triangular faces. Required: the number 
of (27+ m+ 1)-edra that can be constructed from it by the 
process of Art. 4, to have also w triangles. 
As all the S$ summits are triedral, or have 
