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VII.  On  the  Partitions  of  the  Pi-Pyramid,  being  the  first  class  of  ^-goiious  'K.-edra. 
By  the  Bev.  Thomas  P.  Kiekman,  M.A.,  F.B.S.,  Bector  of  Croft  with  Southworth. 
Eeceived  October  14, — Eead  November  26,  1857. 
I.  As  the  partitions  of  an  r-gon  are  made  by  drawing  diagonals,  so  the  partitions  of  an 
r-ace  are  made  by  drawing  diajpeds,  each  a line  in  two  faces  non-contiguous  about  the 
r-ace.  A partitioned  r-ace  standing  on  a partitioned  r-gon  is  a partitioned  pyramid  of 
r-gonal  base  and  vertex.  I am  about  to  determine  the  number  of  such  partitions  of  this 
r-pyramid,  that  can  be  made  with  K diajDeds  and  k diagonals,  so  that  no  two  partitions 
shall  be  syntypous ; ^.  e.  one  the  repetition  or  the  reflected  image  of  the  other. 
I have  proved  in  a memoir  “ On  Autopolar  Polyedra  ” in  the  Transactions  of  the 
Royal  Society  for  1857,  that  the  problem  of  the  polyedra  reduces  itself  to  the  determi- 
nation of  the  ^-edra  generable  from  the  r-pyramid.  Such  an  ^-edron  is  x-gonous. 
JDef.  An  r-gonous  ^-edral  y-acron  is  one  that  by  vanescence  of  convanescible  and 
evanescible  edges  can  be  reduced  to  the  r-pyramid,  and  cannot  be  so  reduced  to  an 
ampler  pyramid. 
The  definition  of  convanescible  and  evanescible  edges  is  found  at  Article  II.  of  the 
above-mentioned  memoir,  as  follows : — 
An  edge  AB  is  said  to  be  convanescible,  when  neither  A nor  B is  a triangle,  and  AB 
joins  tw’o  summits  which  have  not,  besides  A and  B,  two  faces,  one  in  each  summit, 
collateral,  nor  covertical. 
An  edge  ah  is  said  to  be  evanescible,  when  neither  a nor  b is  a triace,  and  the  two 
faces  about  ab  are  not,  one  in  each,  in  two  summits,  besides  a and  b,  collateral,  nor  in 
one  face. 
II.  Theorem.  No  r-gonous  polyedron  has  an  (r-j-lj-gon  among  its  faces,  nor  an 
(r4-l)-ace  among  its  summits. 
For  if  it  has  an  (r-|-l)-gonal  face,  and  no  vanescible  lines  out  of  that  face,  it  is  an 
(r-l-l)-pyramid,  contrary  to  hypothesis;  and  if  it  has  such  a face  and  such  lines  out  of 
that  face,  these  can  be  made  to  vanish,  until  the  figure  is  an  (r-j-lj-pyramid,  ^.  e.  it  is 
(r-}-l)-gonous ; contrary  to  hypothesis. 
In  the  same  way  the  theorem  is  proved  if  the  figure  has  an  (r-flj-ace. 
Cor.  No  r-gonous  polyedron  can  be  reduced  by  vanescences  (^.  e.  disappearances  of 
convanescible  and  evanescible  edges),  to  one  having  an  (r-l-l)-gon  or  an  (r+lj-ace ; i.  e. 
no  r-gonous  polyedron  contains  an  (r-j-l)-gony. 
III.  The  first  family  of  r-gonous  polyedra  are  those  arising  from  partitioning  the  base 
and  vertex  of  the  r-pyramid,  or,  which  is  the  same  thing,  from  laying  a partitioned 
r-ace  upon  a partitioned  r-gon. 
MDCCCLVIII.  X 
