212 
EEV. T. P. EIEKMA^ ON AUTOPOLAE POLTEDEA. 
partition, and the ^-partitions of a polygon, a pencil and a pyramid, will probably be 
found still more inaccessible than our new and unamiable acquaintances, the commensu- 
rahle ^-partitions of a line. 
XXXVI. It may be useful to give a list of the autopolar w-edra as far as 71 = 8 , which 
are not pyramids. 
The only such autopolar hexaedron is obtained by drawing 13 in the 4 -diagonal Q. 
dividing it into the triangles 6 and 5 . Its ten edges are 
4 i 343 o 36 ^2434336 4332^435 ^ 643 ^ 54 i 
41^432^6 82434330 43323435 344 i 4 i 35 364385415 
where the heavier type denotes the faces, and the lighter the summits. The leading 
system are the final pair of gamics, of which the first is convanescible and the second 
evanescible. 
There are three autopolar 7 -edra of the class P (VII.), obtained by drawing the 
generators 13 , 14 and 25 in the pentagonal base Q, all of the class P. These are — 
41353287 
4i353237 
32844337 
82344337 
43433446 
43433440 
34823546 
84323540 
354i4i46 
854i4i40 
3743404i^i 
87434541 
( 13 ) 
4i353246 
^2443346 
^3834446 
44323537 
^54i4i37 
40443 - 41^1 
45443741 J 
( 14 ) 
4i353246 
82448340 
83334440 
44323537 
854i4i37 
3i454237 
3i454237 
4284334:6 
42343340 
33833446 
83338440 
34424546 
84424540 
458i3i37 
453i3i^>7 
364 : 24745 ^* 
85434745 / 
( 25 ) 
the last two of 
which are in 
order the 7 -edra ( 13 ,) and ( 12 ,) of art. XX. They have all 
three quadrilaterals and four triangles. 
Two more of the class S are generated from the 5 -edral pyramid by dra'wing 25 to the 
mid-point of 14 , and 25 to that of 34 ; viz. 
44844237 
42833345 
33423445 
344 i 3546 
^ 54 i 4 i 36 
^7424035 
4 i 344237 
42333340 
83423440 
844 i 3540 
854 i 4 i 30 
874245^5 
^1845245 
33623537 
^5623445 
343 i 3 i 40 
45623785 
3 i 346245 
62338337 
83523537 
85^28440 
843 i 3 i 40 
45503735 
(2. 4 i) 
3 i) 
Of these two, which are identical with ( 23 ') and ( 24 ') of art. XX., the fii-st has three 
quadrilaterals and four triangles, while the second has a pentagon, a quadrilateral, and 
four triangles. In all these 7 -edra 6 and 7 are the faces into which the base Q is 
divided. 
XXXVII. Of autopolar 8 -edra there are five generable from the 7 -edral pyramid, by 
drawing 13 , 15 , 35 , 14 , 36 , in the 6 -gonal Q, all of the class P. 
