200 
EEV. T. P. KIEIQIA^’' OX AUTOPOLATi POLITEDEA. 
summit 3, so that its union with 2, mil give only a 4-ace ; the like remark is to be made 
on the second. Yet nothing but the signatures prevents either of these lines from 
vanishing so as to yield a pentace. Now as these signatures are not essential to any 
polyedron, it is necessary, before we pronounce positively that the s before us (12J 
cannot be generated from the 6-edral pyramid, to satisfy ourselves whether, by any 
rearrangement of the signatm’es, the autopolar character being preserved, this vanescible 
edge 2 cannot be made to stand as the gamic of one not meeting it. 
If we exchange the signatures of the 4-aces 1 and 3, of the triaces and 2, and of 
the triaces 4 and 2^ we have the result follo^ving, still nodally autopolar: — 
Sgd:! 32,82 
3282‘^ i 82, 
^ o 32,4 i 43 
4 o 4:33 o ,84 
4 o 34434 i 
3232,4 o 4 :i 
438430 3o, 
30,4:33480, 
34^34440 
4 i 8232,4 o 
32,4 i 434 o 
3440^330, ’’ 
which is, placing 
the gamics one over the other. 
324 :i ^2,82 
438430,80, 
4 o 433 o ,84 
4 o 34434 i 
'^ o 32,4 i 43 
3232,^0^1 
'^182,3282 
3480,30,4:3 
4380,344:0 
34434 i 4 o 
32,4: i 434 o 
32,4 o 4 i 32 ? 
the heptaedron (P) made from the 6-edral pyramid, which has the two convanescibles 
^2,4:i434o- 
It is thus proved that one (r+Sj-edron s, namely, when ?’=4, must be rejected as 
reducible to the (r+2 = )6-edral pyramid, by the convanescence of an edge not in o. 
Wherefore — is to be added to the number foimd in XIX. 
XXI. Next, let o' be the ^r-ace in As the imited rank of o and o'=7’-j-3. 
o'>>|(r-i-3); then ?/<};-|(r-{-3). Now y4>5, and can be =5 only when there is no con- 
vanescible g, as it has been just proved: therefore ^4>4 and r‘>*5. Then o' is either a 
4-ace or a triace, and a 4-ace only when o=o', in which case it is indifferent which of 
the two is called o. That is, o must be this x-^ce in g wherever x> 3. We are then to 
look in our (7’-j-3)-edron s for a convanescible edge through o. This can be none other 
than the common edge of the only 4-laterals, H and E, which are contiguous only when 
H=E+1. The ?/-ace in g cannot be a summit of O, because g, being in the two faces O 
and E that have the edge OE, could not be convanescible. Therefore the _y-ace is neither 
e noi-y], these being both in O, but is another summit of O', which must be a triace. 
And as x-\-y=i\ie united rank of o and o', of which o is the A’-ace, o' is a triace, and 
O' is a triangle. Therefore g is the edge o(o+I) through (o+l) a summit of the tri- 
angle O', ^. e. through (OEH), i. e. through ^OE(El|lI)j, g bemg E(Eipl). Therefore 
O'E and O'(Eipl) are edges of O'- The signatui-es of the pyramid about e are thus 
exhibited : 
...G-2 G-1 G G+I... 
... : o+I e e—1 : ... 
the colon being the point/. The edges of (o, o-f-l, /) are 00', 0'(G— 1) and 0'(G— 2), 
the two last being O'E and 0'(E-f I); therefore G — 1=E, and e is nodal. Two edges 
