REV. T. P. KIRKMAN ON A CLASS OF POLYEDRA. 
401 
and two others in that of the I sections, which may happen in 2^ different ways. 'J'his 
arrang-ement, c353</e35^, in which cdef are supposed not to be triangles, will then be 
found 2^ times with every disposition of the remaining k — 2 triangles first cut, and 
with every (/— 2) sections of angles not in triangles, of which angles the number is 
x—\—2k. We have therefore counted the completed arrangement ...c353de353f..., 
, t t (ar— 1— ;t) 
in the number 2 
fl+l 
instead of 
i\-i 
2^ .2*‘ 
( 1 - 2 + I 
times, 
2k-2 
(a? — 1 — 2ky~^^~^ 
f/-2 + l 
times. 
The same error has been made with every value of e:^k, and e>»/, or e:t> the least of 
k and /, and this with every set of e twice cut triangles that can be selected out of k. 
Hence there is an error made in excess, in 2*- 
{w — 1 — 
1 
of 
(2'— 1).2''-^- 
(■r— 1 — 2A:)^-^l-‘ ^^1-' 
il—e+ 1 (e + 1 ^ 
for every value of the least of k and I ; for, in supposing e twice cut triangles, 
we assume that e>A:, and that k-{-l—k<^e. The number required in the problem is 
thus proved to be 
(a?— A — !)*■■ 
I/+ 1 
2,.(2*— 1)2*-". 
(a? — 1 — 
l+i-e 
le+\ 
for all positive values of e not greater than either k ov 1: which was to be found. 
This function I shall denote by the symbol k, 1). It expresses the number of 
(j?+/i’+/)-edra that can be made from any .r-edron having an (j?— l)-gonal base and 
k triangular faces, of which no two are contiguous, by removing k-\-l of the summits 
about the base, so that no edge shall be entirely removed, and that no one of the k 
triangles shall remain untouched. Of course k-{-l':if>x—\. Its values are 
ii{x, k, 0) = 2'^; ii{x, k, 1) = 2* \2x—3k — 2) ; 
ii{x, 1, l) = 2 
{x — 2Y~'^ (a?— 3)^“*!“' . 
/7+r 
(I 
ii{x, k, 2) = 2’‘-\l{x-3){x-4:)-k.{x-\-2k)']-3.2’‘-\k.{k-\); 
2, 1 ) =4.fc2^1'-4. <-2=^-3. 
\\+l 
&c. 
n 
&c. 
\l^ 
Let I(j?, k) be the total number of irreversible j:-edra on an x— 1-gonal base that 
have k triangular faces. Then 
l{x, k){ii.(x, k, /)}, part of l{x-\-k-^l, k-\-l), 
is the whole number of from irreversible j-edra 
having k triangles, so as to have k-[-l triangles. Others can be cut to be also part of 
