PARTITIONS OF THE X-ACE AND THE X-GON. 65 
a—2|1 3a—3 | 1 
(+1) he tact) : 
er lm —— “27 
ene f ie y < 
F Sqeaay cal. i i 23 
x R3(62+4+1,3x); or 
IR*(674+m+1,87)=4-R3(6x 41,32) 
r—2|1 
| mie 
a+1 e 
x ee Uy Get eg sate 
Se == Pen 
m 37—3 | 1 m 2Qa—2 | 1 
Gy) m Get) ms) 
a eee 
which is part of the number I(6z+m+1,8z). 
24. We have now all the formule necessary for the 
enumeration of the triedral partitions of the r-ace, that is 
of the triangular partitions of the 7-gon,—— in other words, 
for the enumeration of the (w+ 1)-edra on an w-gonal base 
whose summits are all triedral, or of the (w+ 1)-acra having 
an z-ace and only triangular faces. At least we have all 
the formule except the following (Art. 4), P standing for 
polyedra, 
P(2v+1,z7)=P(v4+1,2)+ P(w+1,3) + P(w + 1,4) 
Mies +P(¢+1,$2),=P(¢+ Ly 
which means that the (22+1)-edra on 2wz-gonal base 
having triedral summits and 2 triangles, which includes 
all the subjects of operation in the preceding problem, are 
in number equal to the (#+1)-edra on wz-gonal base, 
haying triedral summits and two or more triangles, i.e. 
to the (v+1)-edra having an 2-gonal base and triedral 
summits. When 2 is odd, the term P(x+1,}z) is zero, 
the series terminating with P(v+1,}(v-1)). It is evident 
that the reversible or irreversible character of P(~+1,/) 
remains in each figure, when the 2 summits of the base 
are cut away. 
VOL. XV. K 
