BEINQ  THE  EIEST  CLASS  OF  E-OONOHS  X-EDEA. 
155 
in  G with  any  of  the  irreversibles  in  G',  giving  N|.  * N"  and  as  many  bigenerates.  Of 
the  N"  fc  irreversible  partitions  any  one  in  G may  be  combined  with  another  in  Gf,  and 
again  with  that  reversed,  so  that  each  of  the  ^N"  ^(N"  1)  pairs  gives  two  bigenerates. 
For  example,  the  two  bigenerates 
have  each  the  same  irreversible  partition  G on  the  lower  side  in  the  same  position,  and 
each  the  same  irreversible  partition  G'  above,  but  in  positions  one  the  reverse  of  the 
other. 
Hence  the  entire  number  of  bigenerates  is 
where  N'  and  N"  have  the  values  just  found. 
And  this  number  N,.  * is  to  be  subtracted  from  the  results  of  our  enumeration  by  pre- 
ceding formulae,  being  the  exact  number  of  (r-j-K  + l)-acral  (r+^-{-l)-edra  that  were 
twice  constructed,  each  one  Q being  made  both  by  laying  A upon  G,  and  A'  upon  G'. 
XVIII.  Thus  we  have  completely  determined  all  the  partitions  of  the  (r-|-l}-edral 
p^Tamid.  All  that  is  necessary  is  to  give  to  k every  value  from  k—^  to  k—r—'^  in  the 
formulae  of  Art.  XI.  and  the  same  range  of  values  to  K ; then  to  give  to  k in  the  two  pre- 
ceding articles  every  value  from  k=\  to  k=r—^;  for  here  since  neither  k nor  K<1, 
^=r— K— 3>»r— 4. 
And  we  have  at  once  the  number  of  r-gonous  (r+K-[-l)-acral  (r+i^:-f-l)-edra  of 
the  first  class,  by  giving  to  k and  K their  values  in  (XI.)  and  (XVII.).  That  is,  if 
n,.,  K,  k be  Ibis  number, 
K)(R'“^*(r,  ^)+R'“^(r,  k)^W\r,  k)) 
+E^''^"(r,  K)(K'“«'^‘'(r,  ^’)-f  R'*(r,  k))-\-W^‘(r,  K)(E'“^‘''(r,  A:)+E'"«’(r,  kfj 
K) . Pd'"“(r,  /l^) J x li  (2+J])  + K) . E'“^(r,  K) . R*(r,  k) } x hjn 
+ {F(r,  K).R*(r,  k}+W(r,  K).V(r,  k)}j.+2V(r,  K).V(r,  k)j^ 
-(N,.=)(i.K,,(N:,,-I)-fN;,,.N;,+N;,(N; ,-!)); 
T , T T 
where  is  the  greatest  common  measure  of  - and  -,  and  !^A  is  the  greatest  integer  in 
^A,  and  where  R'(r,  k)  denotes  the  entire  number  of  ji’-ly  reversibles  about  all  axes,  the 
rest  of  the  notation  here  used  being  that  of  my  paper  referred  to  in  Art.  V. ; and  where 
Y 2 
