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EEV.  T.  P.  KIEKMAN  ON  THE  PAETITIONS  OF  THE  E-PTEA:inD, 
our  first  position  of  A on  G.  Then,  if  we  reduce  0'  by  an  edge  at  once,  till  it  is  only  a 
single  edge,  we  shall  obtain  0'  different  combined  configurations ; because  no  two  of 
them  show  the  same  distance  measured  in  the  same  direction  between  the  A'  and  G' 
nearest  each  other.  If  we  reduce  0'  to  zero,  we  see  again  G'  under  A',  as  in  our  first 
position,  and  begin  here  a series  of  steps  that  reproduce  the  previous  figures.  Hence 
the  exact  number  of  different  results  is  0^,  the  least  value  of  z in 
a--=.b--\-z. 
3 * — 
the  numbers  a and  h of  intervals  r.j  and  rA  of  whole  edges  from  A'  to  A'  and  fi'om  G' 
to  G'  being  measured  from  ]).  This  z is  the  greatest  common  measui’e  of  r.j  and  r;?, 
i.  e. 
0'=jl^  whole  edges. 
Wherefore  this  is  the  number  of  different  figm’es  obtained  by  laying  one  of  B'lr,  K)  on 
one  of  I‘(r,  ^),  and  the  same  exactly  is  that  obtained  by  laying  one  of  T(r,  K)  on  one  of 
h).  No  new  figures  can  be  obtained  by  reversing  the  irreversible  G,  or  by  turning 
inside  out  the  irreversible  A,  because  A in  the  first,  and  G in  the  second,  of  these  cases 
being  reversible,  we  shall  merely  obtain  by  that  reversal  a reflected  image  of  the  figure 
before  the  reversal. 
XI.  It  remains  that  we  lay  A one  of  T(r,  K)  on  G one  of  I‘(r,  i^),  an  irreversible  on 
an  irreversible. 
A being  laid  in  any  way  on  G,  we  see  at  a certain  point  ^ the  configuration  A"  over 
the  configuration  G".  A"  is  found  in  Kj  times,  and  G"  in  G i times.  The  least  interval 
in  whole  edges  >0  between  an  A"  and  a G"  in  our  first  position  is  the  least  value  of  z in 
T T 
a-=h-r-\-z^ 
3 * — 
r.j  being  the  number  of  rays  in  one  irreversible  sequence  in  A,  and  r.i  that  of  summits 
in  one  irreversible  sequence  in  G.  And  we  have 
r 
for  the  number  of  different  figures,  each  showing  a different  distance  measured  in  one 
direction,  between  the  nearest  A"  and  G".  If  we  now  either  reverse  G,  or  turn  A inside 
out,  we  can  double  this  number  of  results,  for  A in  the  first  case  and  G in  the  second 
being  irreversible,  we  obtain  figures  which  are  not  reflected  images  of  precedmg  ones. 
Wherefore  the  entire  number  of  figures  obtainable  is 
by  laying  any  one  A of  T(r,  K)  upon  any  one  G of  I'(r,  Jc). 
