BEING  THE  EIEST  CLASS  OE  E-GONOUS  X-EDEA. 
151 
Collecting  our  results,  we  have  shown  that  there  are 
ways  of  laying 
A one  of  K)  on  G one  of  k)  or  of  k)^ 
or  A one  of  K)  on  G one  of  k)  or  of  k), 
or  A one  of  K)  on  G one  of  k),  k)  or  k), 
or  A one  of  K)  on  G one  of  R''”‘’(r,  k). 
And  there  are 
different  ways  of  laying 
or 
Also  there  are 
A one  of  K)  on  G one  of  R'^(r,  k), 
A one  of  K)  on  G one  of  k). 
— . different  ways  of  laying 
j\i 
or 
And  there  are 
A any  one  of  E/(r,  K)  on  G any  one  of  I'(r,  k), 
A any  one  of  h(r,  K)  on  G any  one  of  k). 
T 
2 • — different  ways  of  laying 
j\i 
A any  one  of  V{r,  K)  on  G any  one  of  V{r,  k). 
XII.  It  is  certain  that,  in  every  one  P of  these  figures,  the  configuration  with  respect 
to  the  r-gon  1 2 3 . . r,  is  different  from  that  seen  with  respect  to  the  r-gon  1 2 3 . . r upon 
any  other  figure  P'.  But  it  remains  to  be  considered  whether  there  may  not  be  on  P 
another  closed  r-gon,  whose  summits  are  not  1 2 3 . . r,  about  which  is  seen  that  configura- 
tion which  we  read  on  P'  about  the  r-gon  1 2 3.,r.  If  this  be  so,  the  (r+K-{-l)-acral 
(r-}-^+l}-edron  P may  be  merely  the  (r-fK-l-lj-acral  (r-|-^-l-l)-edron  P'.  That  is,  P 
may  be  reducible  by  vanescence  to  two  (r-l-l)-edral  pyramids  not  having  the  same 
signatures,  and  may  be  considered  as  A laid  upon  G,  or  as  A',  a differently  partitioned 
r-ace  from  A,  laid  on  G,  a differently  partitioned  r-gon  from  G. 
If  P has  this  double  character,  I call  it  a higenerate  r-gonous  (l  + K)-acral  {\-\-k)- 
edron ; and  if  it  can  be  made  by  laying  A on  G,  and  by  laying  A'  on  G',  and  by  laying 
A"  on  G",  &c.,  I call  it  a multigenerate. 
XIII.  We  are  thus  compelled  to  inquire,  how  many  multigenerates  of  the  first  class 
can  be  made  by  laying  a (I-f-Kj-partitioned  r-ace  on  a (I-f-Z^j-partitioned  r-gon;  for  as 
we  are  enumerating  only  those  of  the  first  class,  we  have  no  repetitions  to  fear  out  of  it. 
It  is  true  that  some  of  these  (r+K+Ij-acral  (r+^+I)-edra  of  the  first  r-gonous  class, 
are  also  (r-l-K+I)-acral  (r-f^+lj'edra  of  the  second;  as  A'  having  K — e diapeds,  and 
laid  on  G'  having  k diagonals,  and  also  e diapeds  of  the  second  class,  may  give  the  same 
polyedron  with  A having  K diapeds,  laid  on  G having  k diagonals.  This  will  perplex 
the  discussion  of  the  second  class,  but  does  not  trouble  us  here. 
