BEING  THE  EIEST  CLASS  OE  E-GONOUS  X-EDEA. 
149 
we  obtain  both  for  0 even  and  0 odd, 
different  figures, 
whenever  a trace  can  be  laid  upon  an  axis,  0 even  being  finally  reduced  to  zero,  and  0 
odd  to  half  an  edge. 
But  if  A has  only  diachorial  traces  and  G only  diagonal  axes,  or  if  A has  only  acho- 
rial  traces  and  G only  agonal  axes,  we  can  lay  no  trace  upon  an  axis,  if  the  r rays  pass 
through  the  r angles  of  G.  Our  first  position  will  therefore  show  the  0 reckoned  as 
above  diminished  by  half  an  edge,  and  our  last  position  will  show  0 diminished  to  half 
an  edge.  Wherefore  0 must  be  even,  as  is  also  evident  from  the  consideration  that 
the  interval  between  either  two  traces  or  two  axes  in  these  cases  is  an  even  number 
of  half-edges,  so  that  their  greatest  common  measure  is  even.  Hence  the  number  of 
figures  obtainable  by  the  reduction  of  0 is  one  less  than  that  obtained  above  for  0 
even,  ^.  e. 
is  the  number  of  different  figures  attainable,  when  no  trace  can  be  laid  upon  an  axis, 
by  lading  A upon  G in  every  possible  way. 
Now  we  can  always  lay  a trace  upon  an  axis,  unless  either  the  traces  are  all  achorial 
and  the  axes  all  agonal,  or  the  traces  are  all  diachorial  and  the  axes  are  all  diagonal. 
IX.  Hence  we  have  the  Theorem : 
The  number  of  (r+K-|-l)-acral  (r+^+lj-edi-a  that  can  be  made  by  laying 
A one  of  K)  on  G one  of  ^),  or 
A one  of  K)  on  G one  of  Jc), 
is 
and  the  number  of  them  obtainable  in  every  other  case  by  laying  aj-ly  reversible  (1+ /^)- 
partition  A of  the  r-ace  on  an  ^-ly  reversible  (l+^)-partition  G of  the  r-gon  is 
1 ^9,-i.TI 
7“ 
where is  the  greatest  common  measure  of  r.j  and  r.i^  and  is  the  greatest  integer  in 
IN 
2-’  . 
X.  Now  let  A,  a (l-fKj-partition  of  the  r-ace  reversible  about  J axial  planes,  be  laid 
on  G,  an  f-ly  irreversible  (l+^’j-partition  of  the  r-gon. 
If  we  lay  A in  any  way  on  G,  so  that  the  r rays  pass  through  the  r angles  of  G,  we 
see  at  the  termination  of  a certain  trace,  a configuration  G'  in  G under  the  axial  con- 
figuration A'  in  A.  This  A'  is  seen  in  Aj  times,  at  half  the  2j  terminations  of  traces ; 
and  G'  is  seen  in  G ^ times,  at  the  first  point  of  each  of  the  ^ equal  h-reversible  sequences 
that  are  read  round  the  r-gon,  beginning  at  jp.  Let  0'  be  the  number  of  entire  edges 
between  that  A'  and  G that  are  nearest  to  each  other  without  coincidence,,  observed  in 
