BEING  THE  EIEST  CLASS  OE  E-GONOUS  X-EDEA. 
147 
We  have  still  an  account  to  give  of  the  rays  of  A meeting  in  the  other  extremities  of 
the  two  diapeds  joining  A in  the  (r+1  — e)-ace.  There  must  be  at  least  a set  of  two  rays 
at  each  of  those  extremities.  One  ray  in  each  set  will  be  in  each  non-triangular  face 
about  0,  and  therefore  in  a face  about  the  (r+1  — e)-ace;  another  ray  in  each  set,  that 
most  remote  from  will  be  in  a face  about  the  (2+e)-ace,  and  cannot  therefore  meet 
any  edge  of  the  (r+l — 6)-ace,  because  d is  convanescible ; each  ray  must  therefore  pass 
to  a summit  of  the  r-gon  not  occupied  by  the  r — e — 2 diagonals  above  mentioned,  and 
being  rays  of  A,  neither  can  meet  any  other  ray  on  the  r-gon.  But  r-\-\ — e summits 
of  G have  been  shown  to  be  occupied  by  the  edges  of  the  (^+1 — e)-ace,  and  e — 1 more 
by  the  e — 1 rays  meeting  in  the  (2+g)-ace ; therefore  there  are  no  summits  of  the  r-gon 
remaining  to  which  the  two  last  considered  rays  can  pass  from  separate  extremities  of 
the  two  diapeds  meeting  6.  Q.  E.  A. 
Therefore  Q is  not  (r+l)-gonous. 
In  the  same  way  it  can  be  proved  a fortiori  that  Q is  not  (r+l+r')-gonous. 
V.  This  theorem  being  established,  at  a cost  of  words  of  which  I feel  ashamed,  our 
problem  is  reduced  to  the  enumeration  of  the  different  figures  obtainable  by  laying  any 
(l+K)-partitioned  r-ace  A on  any  (l-l->^;)-partitioned  r-gon  G.  But  we  are  to  exclude 
from  our  reckoning  any  figure  P'  which  is  the  reflected  image  of  another,  P ; for  P', 
being  only  P turned  inside  out  through  some  face  supposed  open,  has  the  same  arrange- 
ments and  ranks  of  summits  and  faces  with  P,  i.  e.  is  syntypous  with  P. 
What  follows  will  be  intelligible  to  the  reader  who  has  before  him  my  memoir  “ On 
the  ^-partitions  of  the  r-gon  and  r-ace,”  in  the  Transactions  of  the  Boyal  Society  for 
1857.  These  partitions  can  be  found  by  formulse  there  given. 
VI.  Let  a (1 -f-^)-partition  A of  the  r-ace,  having  ^ axial  planes  of  reversion,  be  laid 
on  a (1+^) -partition  G of  the  r-gon,  having  ^ axes  of  reversion  (vide  the  above  memoir, 
Art.  LXXIII.  and  Theorems  A,  B,  C,  &c.).  I call  the  intersection  of  an  axial  plane 
with  the  r-gon  a trace,  and  by  an  axis  I mean  always  an  axis  of  reversion  of  the 
r-gon  G. 
If  we  can  lay  a trace  upon  an  axis,  the  r rays  of  A passing  through  the  r summits  of 
G,  we  shall  see,  among  the  angles  > 0 at  which  the  remaining  traces  are  inclined  to  the 
axes,  a certain  least  angle  0.  If  x be  the  number  of  half-edges  of  the  r-gon  between 
two  adjacent  traces,  and  y that  between  two  adjacent  axes,  and  2 that  in  the  angle  0, 
0 is  the  least  positive  value  of  z in 
ax=hy^z, 
a and  h being  numbers  of  these  intervals  x and  y measured  in  the  same  direction  from 
the  united  trace  and  axis.  When  x is  prime  to  y,  it  is  well  known  that  z—\\  and 
when  X and  y have  m for  their  greatest  common  measure,  we  have 
m m — 
ax=hy+m. 
X 2 
