REV. T. P. KIRKMAN ON A CLASS OF POLYEDRA. 
409 
RR^(6a’+l, 6k, 1) =^■^■(3x+l, 3A-, k, ^l), RR^(7, 3, \) — 2, 
IR®(6oC+l, 6k, 1) =^{^7(6.r+l, 6k, Z)+3ii(a:+l, k, ^l) 
—u(2x-j-l, 2k, — Zk, \l)), 
IR^(7, 3, 2) =IR=^(7, 3, 1) = IR=*(7, 3, 0) = 1 : 
rR'"(j:+l, k, x—k)-=.6, 
for if all the summits of a reversible are cut, the result is reversible. 
In IT(x, k, 1), rR’”(j:, k, 1), R”R’"(a?, k, 1), the second capital marks the character 
and multiplicity of the a?-edra having k triangles, from which are cut the {x-\-k-{-l)- 
edra having k-\-l triangles, of which the character and multiplicity are denoted by 
the first. 
To show the use of these equations, we can easily by trial verify the following : — 
R^(4, 3) = 1, R^(5, 2) = 1, R(6, 2)=1, 
R(7, 2)=1, R’(7, 3)=1, P(7,2) =1, 
R(8, 2) =2, 1(8,2) =1, 1(8,3) =1, 
R(9, 3)=2, R(9, 2) =2, R"(9, 4) = 1, 
1(9, 3) =3, 1(9, 2) =2, P(9, 2) =2 ; 
then to find the decaedra on a 9-gonal base, with only triedral summits, we first write 
down the classes, 
R(10, 2) = R(8, 2).RR(8, 2, 0) ; 
1(10, 2) =R(8, 2).IR(8, 2, 0) + I(8, 2). 11(8, 2, 0) ; 
1(10, 3) =R^(7, 3).IR^(7, 3, 0)+R(7, 2).IR(7, 2, 1) + P(7, 2).IP(7, 2, 1) ; 
P(10, 3) = RX7, 3).PR^(7, 3, 0) ; 
R(10, 4) = R(6, 2).RR(6, 2, 2) ; 
1(10, 4) =R(6, 2).IR(6, 2, 2). 
That is, by what precedes, 
R(10, 2)=2.m(4, 1, 0) = 2.2 = 4; 
1(10, 2) =2.i{«(8, 2, 0 )-m(4, 1, 0)} 
= 2.1{4-2}=2; 
1(10, 3) =l.i{n(7, 3, 0 )-m( 3, 1, 0)} + l.i«(7, 2, l) + l.i»(7, 2, 1) 
= 1{8_2}+^+^ = 13; 
P(10, 3) = 1.1 = 1 ; 
R(10, 4) = 1.^(3, 1, 1) = 1.(6-5)="1 ; 
1(10, 4) =1.1{27(6, 2, 2)-n(3, 1, 1)} = 1 .i{72-108 + 41-l} = 2. 
