48 
ME.  A.  CAYLEY  ON  THE  PAETITION  OE  NUMBEES. 
We  may  write 
where  711  has  a given  series  of  values  the  same  or  different.  The  indices  not  divisible  by 
a may  be  represented  by  w,  the  other  indices  by  we  have  then 
where  the  number  of  indices  wp  is  equal  to  Jc.  Hence 
= n(l — f e-”')n(l  ; 
or  since  ^ is  a root  of  [1 — = and  therefore  f“=l,  we  have 
and  it  may  be  remarked  that  \.in  = v (mod.  <2),  where  {'<«,  then  instead  of  f we  may 
write  a change  which  may  be  made  at  once,  or  at  the  end  of  the  process  of  develop- 
ment. 
AVe  have  consequently  to  find 
>/f=COeff.  y in  f-'  n(i_g»e-»Qn(l-e-“/'i)' 
The  development  of  a factor  . _ is  at  once  deduced  from  that  of  =<5  and  is  a 
series  of  positive  powers  of  t.  The  development  of  a factor  ^ _g-apt  is  deduced  from  that 
of  contains  a term  involving  y-  Hence  we  have 
n ( 1 — g n ( 1 — e-  “p*) 
_p-apt\ .-1-A_,t  + Ao  + &c., 
and  thence 
The  actual  development,  when  k is  small  (for  instance  ^=1  or  k=2),  is  most  readily 
obtained  by  developing  each  factor  separately  and  taking  the  product.  To  do  this  we 
have 
1 —ce~ 
1 — c (1  — cp  ^(1  — c)^  ^ (1 — cy 
where  by  a general  theorem  for  the  expansion  of  any  function  of  e‘,  the  coefficient  of 
is 
-(-y 1 0/ 
- n/  i-c(i  + A)" 
(where  as  usual  A0-^=H — 0-^,  A®0-^=2-^ — 2.H+0'^?  &^c.)  and 
?: ij-lj-J-  f ^ I ^ 
1— ' 2''  12^  720^^30240^ 
where,  except  the  constant  term,  the  series  contains  odd  powers  only  and  the  coefficient 
of  is  — ; Bi,  B2,  Bj...  denoting  the  series  Bernoulli’s  numbers. 
