146  EEV.  T.  P.  KIEKMAN  ON  THE  PAETITIONS  OE  THE  E-PTEA3nD, 
If  we  draw  k diagonals  in  the  base,  none  crossing  another,  and  suppose  the  base  to  be 
a system  of  ^+1  faces  intersecting  in  those  lines,  we  have  a (r+l)-acral  (r+A’+l)-edron. 
If  we  next  draw  K diapeds,  i.  e.  edges  each  in  two  non-contiguous  faces  about  the 
r-ace,  none  of  them  enclosing  a space,  we  have  K new  summits,  and  have  before  us  a 
(rH-K4-l)-acral  (r+y^;+l)-edron,  of  the  class  now  to  be  considered. 
Another  family  arises  from  partitioning  the  faces  which  intersect  in  the  K diapeds, 
and  the  summits  joined  by  the  k diagonals ; and  a third  family  from  the  partitioning  of 
the  faces  and  summits  upon  the  diapeds  and  diagonals  which  constitute  the  second. 
The  first  class  alone  are  here  to  be  enumerated,  and  the  question  before  us  is  to 
determine  how  many  different  r-gonous  (r+K-|-l)-acral  (r+^’+l)-edra  can  be  made,  by 
laying  a (l  + K)-partitioned  r-ace  upon  a (1+^) -partitioned  ?’-gon,  the  r rays  passing 
through  the  r angles. 
IV.  Theorem.  Every  (r+K-j-lj-acral  Ij'edron  Q,  made  by  la5dng  a (1+K)- 
partitioned  r-ace  A on  a (l+^")"P^i’titioned  r-gon  G,  the  r rays  upon  the  r angles,  is 
/■-gonous. 
For  let  it  be  supposed  that  Q is  (r+lj-gonous : it  is  then  a (r+l-f-Kj-acral 
(r-l-l+^)‘etlron  having  K — 1 diapeds  andi^ — 1 diagonals,  the  vanescence  of  which  ■nill 
reduce  it  to  the  (r+l)-pyramid.  These  K— 1 diapeds  cannot  be  any  K — 1 of  the  K 
diapeds  of  A ; for  of  these  all  the  K must  vanish  to  form  an  r-ace,  much  less  can  K — 1 
vanish  to  form  a (r+l)-ace.  Nor  can  these  diapeds  be  all  of  them  edges  and  diagonals 
of  G,  for  if  G contains  an  (r-|-l)-gonous  system  of  convanescibles,  one  at  least  of  them 
must  be  a diagonal  d,  which  may  be  made  to  vanish  last  of  the  K — 1,  and  must  give 
rise  to  an  (r+lj-ace.  But  d vanishing  can  bring  together  only  four  edges  of  the  r-gon  ; 
it  must  then  bring  together  r — 3 terminations  of  different  diagonals ; but  if  G has  r— 3 
diagonals,  it  is  reduced  to  triangles,  in  which  no  line  is  convanescible ; which  is  contrary 
to  hypothesis. 
Therefore  this  (r+I)-gonous  system  of  convanescibles  must  contain  at  least  one  ray  6 
of  the  r-ace  A ; and  as  convanescibles  may  vanish  in  any  order,  one  by  one,  as  may  dia- 
gonals or  evanescibles,  this  & can  be  made  to  vanish  last  of  the  K— I.  It  must  there- 
fore at  last  carry  at  one  end  an  (r+I  — e)-ace,  and  at  the  other  a (2-l-(?)-ace.  Let 
/3,  the  base  summit  on  6,  be  the  (r+1  — ^)-ace:  this  summit  haAung  only  two  edges  of 
the  r-gon  G and  but  one  ray  6,  has  r — e— 2 diagonals  of  G terminating  in  it;  wherefore 
summits  of  G are  occupied  by  r+I — e edges  of  Q that  meet  at  j(3,  one  of  those  summits. 
The  other  extremity  a of  ^ carries  a (2-l-e)-ace,  the  edges  of  which  are  fu’st, 
secondly,  two  diapeds  of  A,  for  if  at  a meet  only  rays  of  A on  one  or  both  hands,  it 
would  be  in  one  or  two  triangles,  and  would  be  not  convanescible;  thirdly,  e — I rays 
of  A,  because 
2+e_I_2=e-I, 
which  e—1  rays  terminate  at  e~l  summits  of  the  r-gon  G,  all  different  from  the  r-f-I— 
summits  above  mentioned ; for  if  an  edge  of  the  (e-|-2)-ace  meet  one  of  the  (r+I  — ^?)-ace, 
0 would  be  not  convanescible,  by  definition. 
