148 
EEV.  T.  P.  KIEKMAJST  ON  THE  PAETITIONS  OE  THE  E-PTEA^HD, 
Here  m is  the  least  possible  value  of  z’,  for  if  we  say 
ax~hy-^m—n, 
we  shall  have  m—n  divisible  by  m.  Q.E.  A. 
The  number  x is  2r:2ji',  and  y is  2r:2^,  that  of  the  half-edges  of  the  r-gon  divided  by 
the  number  2j  or  2^  of  intervals  between  traces  or  axes.  If  we  put 
r 
for  the  greatest  common  measure  of  r.j  and  this  m is  the  number  of  half-edges  in  0, 
the  least  interval  > 0 between  a trace  and  an  axis. 
Now  let  0 be  diminished  by  an  entire  edge,  every  ray  taking  the  place  of  that  pre- 
ceding it  in  the  dkection  of  revolution  of  the  r-ace.  We  shall  thus  step  by  step  dimi- 
nish 0 either  to  zero  or  to  half  an  edge,  as  m is  even  or  odd.  The  combined  coniigui-a- 
tion  will  be  different  at  every  step,  because  the  least  angle  between  a trace  and  axis  is 
always  diminished,  and  the  configuration  C carried  by  either  end  of  the  revoking  trace 
is  brought  at  each  step  to  stand  over  a different  configuration  m the  r-gon ; for  the 
least  interval  of  the  two  x=r:j  and  y=.ri;  that  is,  no  trace  is  made  to  cross  an  axis  by 
this  process  of  diminishing  0. 
VII.  When  0 has  its  greatest  value,  it  is  either  r.j  or  r:l,  containing  the  whole  inteiwal, 
reckoned  in  half-edges,  between  two  adjacent  traces,  or  adjacent  axes.  If  it  is  even,  it 
can  be  reduced  to  zero,  and  the  last  position  as  well  as  the  first  will  show  a trace  coin- 
cident with  an  axis;  but  whether  these  positions  show  a trace  t over  two  adjacent  axes, 
or  an  axis  a under  two  adjacent  traces,  the  combined  configurations  will  be  different, 
because  the  alternate  axial  configurations  read  at  the  terminations  of  either  traces  or 
axes  are  always  different.  Hence,  when  0 is  reduced  to  zero,  the  figure  is  not  a 
repetition  of  a previous  one. 
But  if  we  diminish  0 below  zero,  we  shall  repeat  in  reversed  order  the  configurations 
seen  when  0 was  >0,  0=^  on  one  side  of  the  united  trace  and  axis  gking  the  reflected 
image  of  the  figure  seen  in  0= — e',  as  everything  is  reversible  about  the  united  trace 
and  axis,  when  0 = 0.  Hence  a trace  is  never  to  cross  an  axis  in  oui’  process. 
VIII.  If  then  we  can  lay  a trace  upon  an  axis  for  a first  position  for  0 undimmished, 
we  shall  obtain  as  many  additional  figures  by  diminishing  0 by  an  edge  at  a step  as  there 
are  entire  edges  in  0.  That  is,  if  0 is  even,  we  get 
I -{- ^>^ = I H-|- = -|  1 2 j 
different  figures ; and  if  0 is  odd, 
different  figures.  Or  if  we  put 
!iN=the  greatest  integer  in  ^N, 
