REV. T. P. KIRKMAN ON A CLASS OF POLYEDRA. 
411 
R(ll, 3) are 4439344535, 5347435535, 4355534636, 3445443737; 
1(11,3) are 5444439353, 5734446353, 6634455353, 7534545353, 
5463447353, 
5553456353, 
5643546353, 
5436444383, 
5436634463, 
6436356344, 
5436453473, 
7346354534, 
6436355435, 
5439344453, 
5438435453, 
5437443653, 
5436444383, 
5436383444, 
5436354435, 
5436374354, 
5534653463, 
6443743643, 
5443743734 , 
5443834634, 
6534643643, 
5534734634, 
5534643734, 
6534734543 , 
5443843553, 
5534653463, 
5534644373 ; 
R( 11, 2) are 4534843544, 44431034444, 5435653454, 6344644364; 
1(11,2) are 4534934444, 6344653454, 6734445443, 6643545443, 
5643545534, 5834444543, 4834444634, 5743544543, 
6444374534, 6553455443; 
r(ll,2) are 5553455534, 7444374443, 6544365443, 6453464534. 
8435445353, 
8436354443, 
5436543563, 
5437535453, 
5443744373, 
6443834543, 
5443753463, 
5734445534, 
4743544634, 
The dodecaedra are found by rather less calculation than the hendecaedra, form- 
ing only eight classes, as follows : — 
1 (12, 5) =R^(7, 3) . IRX7, 3, 2) + F(7, 2 ). IP(7, 2 , 3) + R(7, 2 ). IR(7, 2 , 3), 
R(12, 5) = R(7, 2)RR(7, 2, 3) + R*(7, 3).RR^(7, 3, 2), 
1(12, 4) =1(8, 3)11(8, 3, I) + I(8, 2 ). 11(8, 2, 2) + R(8, 2).IR(8, 2, 2), 
R(12,4) = R(8, 2).RR(8, 2,2), 
1(12,3) =:I(9,3).n(9,3,0) + l(9,2)II(9,2,l) + R(9,3).IR(9,3,0) + R(9,2).IR(9,2,l), 
R(12, 3) = R(9, 3).RR(9, 3, 0) + R(9, 2)RR(9, 2, 1), 
1(12,2) =1(10, 2)11(10, 2, 0)-l-R(10, 2).IR(10, 2, 0), 
R(12, 2) = R(10, 2).RR(10, 2, 0). 
These are, by what precedes, all given numbers ; and, by continuing the process, we 
can finally obtain all the .r-edra on an {x— l)-gonal base, numbered in their proper 
classes, which have only triedral summits. 
I have generalized the expressions Fl”(.r, k, /), R”R'"(a?, k, /), rR”(j7, k, 1), which 
the theory requires for enumerating all the j:-edra having an {x — l)-gonal base, and 
any summits whatever; but the formulae are not worth producing. The number of 
distinctions to be made is too great to be of any ready use. If the a?-edra having an 
(x — l)-gonal base were classed and enumerated according to their summits, it would 
be possible to count all the (a:+/i)-edra on the same (a;— l)-gonal base, by removing 
summits not in the base, thus producing crown-faces, and by the vanishing of edges 
about the crowns and base, thus producing faces contiguous, but not collateral. That 
is, it would be possible to enumerate and classify the N-edra. 
