64. REV. T. P. KIRKMAN ON THE 
points for triply irreversible results. This can be done in 
Mm Qv—2] 1 
(5 +1) i" 
parte 5B! te pe, Qs 
12 |1 
ways, which operations are to be twice again performed in 
the same order and direction in the other two thirds of 
the circuit of our subject. This gives us the result, 
2x—2 | 1 
+1) on 
PR°(62+m+1,32)= aan i -R(6z+ 1,32), 
a portion of I’(672+m+1,32). 
23. Prozsiem o. To determine the number of singly 
irreversible (62+4+m-+1)-edra having 32 triangles that are 
reducible (Art. 3) to triply irreversible (6x +41)-edra with 
3a triangles. 
If on the 62-3 edges k we deposit m points 7, we obtain 
(at 1s 
yoe—4 11 “2 
results. Among these will occur every triply reversible of 
the number R*R*(6z+m+1,3z) (Art. 20) once, and once 
only ; every reversible of RR*(6z+m+1,32) (Art. 21) will 
be found three times, being constructed about each of 
the bisecting axial planes of the subject; every figure of 
PR*(62-+m-+1,32) of the preceding article will present 
itself twice, i.e. each figure and its reflected image once; 
and every singly irreversible here required will be seen 
six times, each figure and its reflection being constructed 
to have the same complete circuit begun at three different 
points of the triple subject. 
From the entire results set down above we must, there- 
fore, subtract the R*R’, three times the RR*, and twice 
the I?R' of Art. 20, 21, 22, and divide the remainder by 
six. This gives us 
1 6x—4 | 1 
IR*(6x +m +1,8x)= risa “gm 
