214 
EEV. T. P. ETEKMAi^ ON AnOPOLAE POLTEDEA. 
There are yet three autopolar S-edra not included in the classes discussed in this 
memoir, being (r+4)-edra generable by gamic pairs from the (r+l)-edral pyramid. 
The first is obtained by drawing in the 4-gonal Q, 42, and then 45 to the middle of 12 ; 
the second by drawing 24 and then 38 to the middle of 24 ; the third by jo ining 65, in 
the mid-points of 14 and 12, as follows : — 
31643536 
81643536 
65644237 
85644367 
43833333 
43633363 
63436433 
83426468 
643 i 3 i 36 
64613165 
6035^764 
80653764 
674068641 
' - ® * ( 42 . 411 ) 
374 . 3 s 64 r 
3 i 444245 
43434336 
43434437 
66436733 
67444533 
45436533 
443 iS 445 V 83 , 82 . 
44 Si 3 i 45 j S-i) 
81444345 
43434365 
43424467 
80433768 
37444565 
45433065 
4 i 44^537 
41443567 
65446253 
854433^3 
63836353 
83633363 
63334453 
33624468 
44416553 
444 i 3668 
664 i 4 i 37 
864 i 4 i 67 
67356830 1 
0 „ . 0 HUH) 
O 7 ^ 508 ^ 6 j 
Thus we find that there are one autopolar 6-edron, five autopolar 7-edra, and fifteen 
autopolar 8-edra, besides the three pyramids. 
XXXIX. We may, for example, effect the reduction of the three last written 8-edra 
to the 5-edi’al pyramid. The only vanescible pair in the first is the seventh pair. As 
it stands, the summits and their signatures read thus, 
31423364^35353738 : 
after the disappearance of that pair, which unites the seventh and eighth summits and 
degrades the second and fourth, they read 
^1^233^4^5^64:7 8* 
The second pair are now vanescible and next vanish, and we read thus, 
3 i 42 533443537 8 , 
after which the vanishing of the sixth pair gives us 
3 i 32533344578, 
the pyramid on the base 1234. 
But if we do not insist on the vanescence of gamic pairs, and on an autopolar result, 
we can reduce the 8-edron readily to the 7-edi’al pyramid, by the evanescence of3T4:23864- 
which makes the face 473 ? then the convanescence of 33426443 5 but the fii'st of 
these two steps gives a heteropolar result. Generally any a’-edron that has at most a 
p-ace or a^-gon can be reduced by single vanescences to the (^+l)‘edral pyramid. 
In the second of the above three 8-edra, the first pair, though at fii’st sight it appears 
a vanescible pair, is not so. For putting s„ and/^ for the summit and face signed n, we 
see in the eighth and second pairs, or the nodal pans, Sj in/,, and S 3 in /, S 3 in 
S 3 in/ 3 . If iiow 3 1444345 evanesces, we have/,=/ 2 , whence, s^ is in / through S 3 , and S 3 
in/ through so that 3 1 44 4 345 is no longer convanescible, / and/ through and s^ 
being covertical, and in fact collateral faces. 
