[ 47  ] 
IV.  Supplementary  Besearches  on  the  Partition  of  Numbers. 
By  Arthue  Cayley,  Esq.,  F.B.S. 
Eeeeived  March  19, — Eead  June  18,  1857. 
The  general  formula  given  at  the  conclusion  of  my  memair,  “ Researches  on  the  Parti- 
tion of  Numbers*',”  is  somewhat  different  from  the  corresponding  formula  of  ProfessoY 
Sylvester  f,  and  leads  more  directly  to  the  actual  expression  for  the  number  of  parti- 
tions, in  the  form  made  use  of  in  my  memoir ; to  complete  my  former  researches,  I pro- 
pose to  explain  the  mode  of  obtaining  from  the  formula  the  expression  for  the  number 
of  partitions. 
The  formula  referred  to  is  as  follows,  viz.  if  ^ be  a rational  fraction,  the  denominator 
of  which  is  made  up  of  factors  (the  same  or  different)  of  the  form  and  if  « is  a 
dhisor  of  one  or  more  of  the  indices  m,  and  k is  the  number  of  indices  of  which  it  is  a 
divisor,  then 
where 
g — X 
1 , - &x 
n(s-l/'^^^)  [1 
^£>=coeff.  - in  if 
/(ge-0  ’ 
in  which  formula  [1 — denotes  the  irreducible  factor  of  1 — that  is,  the  factor 
which  equated  to  zero  gives  the  piime  roots,  and  f is  a root  of  the  equation  [1— .r"]  = 0; 
the  summation  of  course  extends  to  all  the  roots  of  the  equation.  The  index  s extends 
from  s=l  to  s=k;  and  we  have  then  the  portion  of  the  fraction  depending  on  the 
denominator  [1—^®].  In  the  partition  of  numbers,  we  have  (px=l,  and  the  formula 
becomes  therefore 
Xg 
where 
5x 
[i-x^y 
g 
/{ge-f 
* Philosophical  Transactions,  t.  cxlvi.  p.  127  (1856). 
t Professor  Stlyestee’s  researches  are  published  in  the  Quarterly  Mathematical  Journal,  t.  i.  p.  141; 
there  are  some  numerical  errors  in  his  value  of  P (1, 2, 3, 4, 5, 6)  q. 
