158 
EEV.  T.  P.  KIEKMAN  ON  THE  PAETITIONS  OP  THE  E-PTEA:iin). 
thus  by  using  the  circles  12345678,  the  closed  octagon  through  the  summits  of  the 
6-edron,  and  123456,  the  hexagon  on  the  faces: — 
2 
h 
j 
9 
71 
• 
• 
e 
/ 
1 
j 
a 
3 /i 
/ 1 
, 
i 
c d 
e 
8 

a 
b 
c 
h . 
• 
k 
b 
. 
i 
f 
[/ 
9 
a 
• 
k 
1 
1 
/ 
i 
) 
This  arises  from  partitioning  the  only  tesserace  in  the  preceding  T-acron,  and  is  reduced 
to  that  by  the  convanescence  of  any  one  of  its  twelve  edges.  It  is  also  of  the  second 
class. 
The  eighth  is  an  autopolar  heptaedron,- 
'^i354236  434:43333  4^4:23587 
35423641  42353744 
3644^>74i  ^7354i4i  42834486 
44874183  354i4jl37  334:430^2 
e . . . E A d 
. . B c A F . 
. f b B . . . 
. ^ F . . d C 
a C . . . . € 
D « . / . . . 
E . . D c . 
It  is  not  possible  to  reduce  any  of  these  eight  polyedra  to  the  5-based  pyi’amid ; nor  can 
any  diagonal  or  diaped  be  drawn  in  any  of  them  which  shall  not  either  produce  another 
of  the  eight,  or  introduce  di  pentagony,  i.  e.  a 5-gonous  system  of  vanescibles.  There  are 
no  4-gonous  ^r-edra  of  a class  beyond  the  second. 
XXI.  There  is  no  difficulty  in  finding  tentatively  the  number  of  5-gonous  polyedra. 
by  partitioning  the  faces  and  summits  of  the  first  class  of  them,  taking  care  to  introduce 
no  hexagony.  The  partitions  of  the  5-based  pyramid,  that  is,  the  whole  of  tliis  first  class, 
are  given  by  the  formula  of  XVIII.,  in  which 
i.  e. 
n;,,=o  n';,.=i,  n;,,=i,  n;%=o, 
N,,,=0=N,,,. 
And  the  only  partitions  of  the  5-gon  are  R^'”''(5,  0)  = 1,  R™°(5,  I)  = l,  R”"’(5,  2)  = I. 
Wherefore 
