REV. T. P. KIRKMAN ON A CLASS OF POLYEORA. 
40{) 
If we cut a summit of either triangle of the pentaedron, we obtain the system 35344 
about the pentagonal base. This is a reversible hexaedron. 
By cutting the first triangle of this with the pentagon and the second with the 
adjoining quadrilateral, we obtain the system 4364354, which I call an irreversible 
octaedron ; it does not read backwards and forwards the same. 
The double reversible 3434, by one section of its triangles, becomes 435435, a 
double ij'reversible, and by another, 436344, a reversible heptaedron. 
And the system 353535 gives, by cutting the triangles in one way, the irreversible 
437346345, and, in another way, the trebly irreversible 436436436. 
These six varieties comprise all the polyedra that have only triedral summits. 
They are all irreversible, reversible, doubly irreversible, doubly reversible, trebly irre- 
versible, or trebly reversible. 
For if any polyedron exhibited in the faces about the base the fourfold repetition 
of any period of m faees containing k triangles, the vanishing of the Ak triangles 
would give rise to a fourfold repetition of a period of m—k faces, which would con- 
tain k' triangles, k' being not more than k-, and this system around the base would 
reduce, by the vanishing of the Ak’ triangles in it, to a fourfold repetition of a period 
of m — k — k' faces ; and we should obtain at last a base of 4(m— K) sides, admitting 
no further reduction by the vanishing of triangles, i. e. we should obtain a pyramid 
having a tetraedral summit; which is impossible. 
Problem. — An A?-et!ron P being given on an 1-gonal base, and having k triangular 
faces, it is required to determine how many (x+^+O-^dra can be cut from it by the 
removal of k-\-l base summits, so that none of the k triangles shall remain untouched, 
and so that no (j?+A-+/)-edron shall be the reflected image of any other. 
First, let P have about its base a series of x faces which read differently both 
backwards and forwards from every face, i. e. let it be an irreversible a7-edron. 
As no two contiguous faces about the base can be triangles, if x>4 (for then 
there would be a teti’aedral summit at least, at one extremity of their common side), 
1), and k-\-l'^x— J. 
We are bound to cut each of the k triangles once, which can be done in 2* ways, 
giving 2^ different irreversible arrangements of 1 faces. Next we have to cut 
I of the remaining x—\—k angles about the base of P. These may be any I out of 
r— 1— /r, and this gives us 2^^. — arrangements of our sections. But 
these will not be all different arrangements. Any one of them will contain e cases 
of twice cut triangles of P, thus made into e pentagons, and consequently l—e 
triangles which are not cut from triangles. 
Let us suppose e=2, the case in which two pentagons are newly made, standing thus, 
...c353c?e353/... 
Of these four triangles, two were introduced in the distribution of our k sections. 
