ME.  A.  CAYLEY  ON  THE  PAETITION  OF  NUMBERS. 
49 
But  when  k is  larger,  it  is  convenient  to  obtain  the  development  of  the  fraction  from 
that  of  the  logarithm,  the  logarithm  of  the  fraction  being  equal  to  the  sum  of  the 
logarithms  of  the  simple  factors,  and  these  being  found  by  means  of  the  formulse 
log 
;=  log 
I C_  . I c _ c + c^  c + 4c^  + c^  i!.  4- 
1-c  1-c  “*"(1-0)2  2 (i_c)3  6“*“  (]-c)4  24“* 
log—^: — - = — log^+-^ &C. 
S ^2  24  ^2880  181440  ^ 
The  fraction  is  thus  expressed  in  the  form 
1 2 — 
and  by  developing  the  exponential  we  obtain,  as  before,  the  series  commencing  with 
Resuming  now  the  formula 
which  gives  as  a function  of  we  have 
but  this  equation  gives 
and  we  have 
^ ,=s-xi-; 
[1  — a?®]  q—X 
[1  — = (.r — g>)(^ — g*®") . . — g*"" ) 
if  1,  •••  s-re  the  integers  less  than  'a  and  prime  to  it  (a  is  of  course  the  degree  of 
[1— 4“]).  Hence 
. -1 
and  therefore 
. ^ . • ^f= ; 
or  putting  lor  yp  its  value 
where  a is  the  degree  of  [1— and  denotes  in  succession  the  integers  (exclusive  of 
unity)  less  than  a and  prime  to  it.  The  function  on  the  right-hand,  by  means  of  the 
equation  [1  — f'']  = 0,  may  be  reduced  to  an  integral  function  of  f of  the  degree  a — 1, 
and  then  by  simply  changing  ^ into  x we  have  the  required  function  Gx.  The  fraction 
can  then  by  multiplication  of  the  terms  by  the  proper  factor  be  reduced  to  a 
fraction  with  the  denominator  1 — 4“,  and  the  coefficients  of  the  numerator  of  this 
fraction  are  the  coefficients  of  the  corresponding  prime  circulator  ( )pcr  a^. 
Thus,  let  it  be  required  to  find  the  terms  depending  on  the  denominator  [1—^®]  in 
MDCCCLVIII. 
^ 1 . 
(1— «;)(!— a?2)(i—a;3j(l_^4j(l—a;^)(l_a;6j  ’ 
H 
