BEING  THE  EIEST  CLASS  OE  E-GONOIJS  X-EDEA, 
5y 
Such  a figure  Q can  only  be  bigenerate ; for  suppose  that  it  was  trigenerate,  made  by 
laying  A on  G,  or  A'  on  G',  or  A"  on  G".  Then  it 
is  proved  that  A and  A"  have  common 
rays,  and  that  the  only  diapeds  of  A are  hc^  cd 
hence  the  rays  common  to  A and  A"  must  be  ai,  hh, 
eg,  df and  the  figure  must  be  of  the  form 
But  this  is  not  properly  multigenerate,  the  three 
10-gons  jiJigfedcba,  jihgfedpjbfl^,  and  jabcdedppfi^,  being  all  the  same  partition  G,  sur- 
mounted by  the  same  partitioned  10-ace  A.  A 4-generate  Q is  here  entirely  out  of 
question ; for  the  figm-e  must  reduce,  by  the  convanescence  of  ther — Jc — 2 diapeds  of  A, 
to  a simple  r-ace  standing  on  G having  k diagonals.  Wherefore  the  Q to  be  considered 
here  is  bigenerate  only. 
XV.  We  shall  know  the  number  of  bigenerates,  if  we  determine  in  how  many  different 
ways  these  k diagonals  drawn  from  k consecutive  summits  of  the  r-gon  can  terminate  at 
the  T—k — 2 summits  ah  c d.  But  if  G and  G'  had  the  same  arrangement  of  their  k 
diagonals,  or  if  one  were  merely  the  refiected  image  of  the  other,  we  should  have  a 
figure  generated  by  A upon  G only,  ^.  e.  a figure  that  we  have  constructed  and  counted 
only  once  among  those  made  by  laying  (K-j-l)'partitioned  r-aces  upon  (^4-1  j-partitioned 
r-gons.  For  example,  neither  of  the  two  figures  following 
has  been  twice  enumerated ; for  whether  we  take  ah,  he,  and  cd,  or  ap^,  hp^  and  epL^  for 
the  diapeds  in  either,  we  find  indeed  A on  G and  A'  on  G',  but  A differs  in  no  respect 
from  A',  nor  G from  G',  nor  is  there  any  difference  in  the  way  of  applying  the  A to 
the  G,  Either  of  them  is  made  by  laying  in  one  way  only  the  4-parti tioned  10-ace  A 
on  the  5-partitioned  10-gon  G,  here  given  separately. 
Ihe  (1-f  ^)-partitions  now  to  be  enumerated  may  be  either  reversible  or  irreversible. 
If  k is  odd  and  — k — 2 is  odd  also,  that  is,  if  r is  even  and  k is  odd,  one  of  the  k 
diagonals  may  be  a diameter  about  which  the  partition  is  reversible.  If  k is  even,  and 
o,  i.  e.  r , is  either  even  or  odd,  the  partition  may  be  reversible  about  either  an  agonal 
MDCCCLVIII.  Y 
