156 
EEV.  T.  P.  KIRKM.\^  OX  THE  PAETITIOXS  OF  THE  E-PTEA^IID, 
and 
N'  =2i2  a(^-^-2)rM  , o o 
n;.=. 
+ 1) 
, f(»-*-2)*"  1 
STT  -"'■*/• 
And  the  entire  number  £3^  of  partitions  of  the  (r+l)-edral  pyramid,  made  by  parti- 
tioning both  the  vertex  and  the  base  by  diapeds  and  diagonals,  is 
Sc— 2k2aI1,.^  k,  *? 
the  double  integral  being  taken  for  every  value  both  of  K and  k firom  zero  to  r=3. 
The  quantity  subtracted  in  11^,  k,*?  vanishes  for  k=0  and  for  Ic=7' — 3. 
XIX.  Thus  far  the  theory  of  the  polyedra  has  been  opened  and  discussed  without 
descending  to  any  classification  according  to  the  ranks  of  the  faces  and  summits.  We 
have  had  nothing  but  an  r-ace  and  an  r-gon  to  partition.  But  1 do  not  see  how  the 
second  and  higher  classes  of  r-gonous  ^-edra  can  be  enumerated  without  such  a classi- 
fication. This  Avill,  I fear,  introduce  a boundless  complexity,  and  go  far  to  deprive  the 
investigation  of  all  claim  to  scientific  generality.  Yet  others  may  find  out  a more  prac- 
ticable method  of  attacking  this  interesting  problem,  and  I may  live  to  see  the  remain- 
ing cases  of  it  discussed  within  a reasonable  compass.  I wish  the  analyst  joy  of  his 
task  who  shall  undertake  to  complete  what  I have  had  the  good  fortime  to  begin. 
XX.  I have  no  doubt  that  the  number  of  r-gonous  ^-edra  is  always  limited ; but  the 
maximum  number  of  their  edges  is  no  simple  function  of  r.  It  is  worth  while  to  wuite 
out  all  the  4-gonous  polyedra. 
These  are,  first  and  second, — 
32446333  3^42354:5  3g42^63l  0g440i3l 
353i3]^45  3q423344  33333445? 
a 6-acral  5-edron,  and  its  polar  syntyp, — 
3i443245  32^48333  83428433  34428545  3542363i 
36443i3i  353i3i45  33423344  82338445. 
The  heavy  type  expresses  the  faces,  and  the  lighter  the  summits. 
Both  are  thus  represented  in  one  paradigm,  by  the  method  explained  in  my  paper 
“ On  the  Representation  of  Polyedra,”  in  the  Philosophical  Transactions  for  1856,  using 
the  circles  123456  and  12345,  and  writing  a at  .(l,  4)  and  (2,  5),  &c. : 
