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EEV.  T.  P.  KIEKMAN  ON  THE  PAETITIONS  OF  THE  E-PYEA3nD, 
Let  Q be  such  a multigenerate,  made  by  lajdng  A on  G,  and  by  A'  on  G',  &c.  As  A 
is  not  A,  the  r rays  of  A'  will  not  comprise  all  those  of  A ; nor  can  they  be  all  diapeds  of 
A,  for  A cannot  have  more  than  r — 3 diapeds ; nor  can  they  all  be  found  among  the  sides 
and  diagonals  of  G,  for  it  is  impossible  to  make  above  r— 1 edges  and  diagonals  by  any 
arrangement  or  convanescences  to  meet  at  one  summit  of  G ; while  A'  is  reducible  by  con- 
vanescence  to  an  r-ace.  Wherefore  >0,  <r)  of  the  rays  of  A must  be  also  rays  of  A'. 
Let  these  g common  rays  be  aa^,  hb^,  dd^, the  points  abed ...  being  summits  of 
G',  and  a^b^Cidi . . . summits  of  G.  If  the  K diapeds  of  A convanesce,  A becomes  a simple 
r-ace  E,  standing  on  G the  r-gon  a^b^c^d^...,  wherefore  ab,  be.,  cd,  ...  are  diapeds  of  A ; 
and  in  like  manner  bp^,  c^d^ . . . are  diapeds  of  A'.  "V^Tien  A is  reduced  to  the  simple 
r-ace  K,  there  are  no  summits  of  the  figure  on  the  side  of  the  r-gon  b^  c,  d^...  remote 
from  E,  since  A is  laid  on  the  partitioned  r-gon  a^b^c^d^...;  and  if  A be  reduced  to  the 
simple  r-ace  E',  there  are  no  summits  in  the  figure  on  the  side  of  the  r-gon  abed... 
remote  from  E'.  Hence  Q,  the  figure  A upon  G,  is  of  this  form,  for  the  case  r=10. 
Here  G is  the  10-gon  a^bpplpfghij, 
G' hih.el^-gon  abedef  g hi  j; 
The  g common  rays  are  aa,  bb^  cc^  ddp  (^=4) 
The  diapeds  of  A are  ab,  be,  ed ; 
The  diapeds  of  A are  ap^  bp^  e^dp. 
The  diagonals  of  G are  ap,  a]i,  bp,  e^f,  all  rays 
of  A; 
The  diagonals  of  G'  are  ai,  eh,  eg,  df,  all  rays 
of  A. 
The  number  of  these  diagonals,  along  with  the  g common  rays  and  the  lines  aj  and  de, 
must  make  up  in  A the  r rays,  i.  e. 
Jc=T  — g~2, 
of  which  one  must  pass  through  each  of  the  r—g—2  summits  of  G’  fg  hi,  which  are 
also  summits  of  G through  which  pass  the  rays  of  A,  ai,  eh,  eg,  df.  The  number  of 
diapeds  in  either  A or  A is 
K=^-l=r-A:-3. 
XIV.  The  condition  to  be  fulfilled  in  drawing  the  diagonals  of  G is,  that  all  the  e— 1 
lines  ab,  be,  ed  shall  be  diapeds,  i.  e.  none  of  them  in  a triangle,  and  the  diagonals  of  G' 
must  be  so  drawn  that  all  the  lines  ap,  bp^  epL^  shall  be  also  convanescible.  If  these 
conditions  are  fulfilled,  the  figure  is  bigenerate,  othermse  it  is  not.  This  condition  is 
fulfilled,  if  the  k rays  of  A from  ^ h gf  be  drawn  in  any  manner  not  crossing  one  another 
to  one  or  more  of  the  g summits  abed,  and  if  the  k rays  of  A'  from  i h gf  are  drawn  in 
any  manner  not  crossing  one  another  to  any  of  the  summits  a^  b^  e^  d^.  For  example,  the 
rays  of  A,  or  rather  the  diagonals  of  G',  may  be  drawn  all  to  a,  viz.  ai,  ah,  ag,  af\  the 
lines  ah,  be,  ed  will  be  convanescible,  being  three  sides  of  an  open  6-gon  abed  ef;  and 
thus  it  is  easily  seen  that  these  k diagonals  of  G'  through  i h g and  f may  be  distributed 
in  any  manner  upon  the  summits  abe d. 
