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EEV.  T.  P.  KIEOIAN  ON  THE  PAETITIONS  OE  THE  E-PTEA3irD, 
or  a monogonal  axis ; \k  diagonals  being  similarly  drawn  on  either  side  of  it.  But  if  Tc 
is  odd  and  r — k—1  is  even,  e.  if  r be  odd,  no  axis  but  a monogonal  can  be  drawm 
through  the  central  one  of  the  Tc  summits  ^,  h,  g,  &c.,  which  axis  cannot  be  a diagonal. 
Hence  if  k is  even  and  r is  odd,  the  partition  cannot  be  reversible. 
XVI.  First,  let  k be  odd  and  r even.  We  can  draw  a diameter  ^ through  the  central 
point  of  ^,  A,  g,  &c.  for  one  diagonal,  and  can  draw  ^(k—1)  on  one  side  of  ^ to  termi- 
nate in  one  or  more  of  \{r—k — 1)  summits,  viz.  the  central  point  of  ahc ...  and  those 
on  one  side  of  it,  in 
different  ways,  and  the  figure  can  be  completed  symmetrically  about  S in  so  many  ways 
into  a reversible  (l-}-A)-partition. 
If  k is  even  and  r is  even,  we  can  draw  \k  of  the  diagonals  to  one  or  more  of  the 
corresponding  half  of  the  r — A— 2 summits  ^,  A,  y,  &c.  in 
A (t  — ]c  — 
^ r 1-7-7-  -4 — different  w’^ays, 
and  complete  the  figure  by  drawing  the  remaining  ^k  into  a (1  +A)-partition  reversible 
about  an  agonal  axis  having  on  each  side  of  it  \k  diagonals. 
If  k is  even  and  r is  odd,  we  can  draw  ^k  of  the  diagonals  to  the  — k — 1)  points 
consisting  of  the  central  point  of  the  r — k — 2,  and  the  rest  on  one  side  of  it,  and  the 
figure  can  be  completed  symmetrically  about  the  monogonal  axis  through  that  central 
point  into  a reversible  (I-l-A)-partition,  in 
\ — different  ways. 
Wherefore  the  entire  number  of  reversible  ways  of  drawing  k diagonals  from  k conse- 
cutive summits  f,  h,  g ...  to  one  or  more  of  the  [r — k — 2)  summits  a,  h,  c ...,  is 
4*1 1 
The  entire  number  of  ways  in  which  k diagonals  can  be  drawn  from  the  k points 
i,  A,  g,  ...  to  one  or  more  of  the  r—k—2  points  ahc....,  is 
(r— A:— 2)^1’ 
f+T 
among  these  every  reversible  partition  comes  once  only,  but  every  ii-reversible  Q occurs 
twice;  for  Q and  its  reflected  image  both  occur.  Hence  the  number  of  hreversible 
(I-l-A)-partitions  is 
N" 
/jqr[ 
XVII.  Of  these  * reversible  modes  of  partition  any  one  can  be  combined  in  G with 
any  other  in  G',  giving  ^N'.  i.(N^ — I)  pahs,  and  as  many  bigenerates,  which  have  been 
each  twice  constructed  and  counted.  Of  these  * modes  any  one  may  be  combined 
