REFERENCE TO THE ENUNCIATION OF POLYHEDRA. 
43 
[308 
process might 
3 to a number 
lerived from a 
f 2/ 3 + y 3 + &c.)- 
ms which are 
fie, and so on; 
every m-wise 
it is derivable 
and the sum 
the application 
, in which the 
he number of 
il summit, and 
lis notation is 
are two forms 
gures shows at 
. that derived 
igle 456, while 
bs stands upon 
the polyacrons 
lerable number 
heptacrons will 
, 555, the first 
lore than two 
and 446, and 
two tripleural 
3S gives there- 
356, 456 and 
,cron has two 
er the second 
308] 
4. The heptacron 3444555 has four kinds of faces, viz. 355, 455, 445 and 444, 
and the first process gives therefore 4 octacrons. The heptacron has one tripleural 
summit, and the basic quadrangles 3545 which belong to it are of the same kind ; 
the second process gives therefore 1 octacron. 
5. The heptacron 4444455 has only one kind of face, viz. 445, and the first 
process gives therefore 1 octacron. There are two kinds of basic quadrangles, viz. 4545 
and 4445, and the second process gives therefore 2 octacrons. 
The number of octacrons would thus be 20, but by passing back from the octacrons 
to the heptacrons, it is found that there are in fact only 14 octacrons. Thus the 
octacron 33336666 has only one kind of tripleural summit 666 (the summit is here 
indicated by the symbol of the basic polygon) and the octacron is thus seen to be 
derivable from a single heptacron only, viz. the heptacron 3335556 from which it was 
in fact derived. But the octacron 33345567 has three kinds of tripleural summits, viz. 
567, 557 and 467, and it is consequently derivable from three heptacrons, viz. the 
heptacrons 3335556, 3344466 and 3344555, and so on. The passage to the heptacrons 
from an octacron with one or more tripleural summits is of course always by the 
first process, but for the last two octacrons, which have no tripleural summits, the 
passage back to the heptacrons is by the second process: thus for the octacron 
44445555 we have but one kind of tetrapleural summit 4555; but as opposite pairs 
of summits of the basic quadrangle are of different kinds, viz. 45 and 55, we obtain 
two heptacrons, viz. 3444555 and 4444455. The octacron 44444466 has but one kind 
of tetrapleural summit, viz. 4646, and the pairs of opposite summits of the basic 
quadrangle being of the same kind 46, we obtain from it only the heptacron 4444455. 
It may be remarked that for the five heptacrons respectively the values of the 
sum y x + \y* + ly-i + • • • are 
l+i+i, i + i + i + i, £ + £ + i + i + i, l + i + i + i+i, l + i + b 
giving for 2 (y x + \y 2 + ^y 3 + ...) the value 14, as it should do. 
;he faces denoted 
are faces of the 
6—2