ME.  A.  CAYLEY  ON  THE  PAETITION  OE  NUMBEES. 
51 
whence,  finally, 
^.f=i(42+23f),  4,*=3h(42+23a-): 
and  the  partial  fraction  is 
1 42  + 
which  is 
^ 42-19a7-23a;2 
*3. > 
and  gives  rise  to  the  prime  circulator  ^§'(42,  —19,  —23)  per  3,. 
The  part  depending  on  the  denominator  1— is 
A_.  . 1 
where 
We  have  here 
(1  — e“5(l 
=A_6^+A_5-^---+A_,yd- &c. 
log ^+2^  2880^  ~ 
and  thence  the  fraction  is 
which  is  equal  to 
1 &e. 
„ (>2  24  576 
720  ^ 
3087  64827  1361367  .,  , 
o-r  ^r,c  I -\-  J280  ^ ”■ 
128 
1 1 , 7 1 , 77  1 , 245  1,  43981  1 199577  1 
'ron  y6;“l“ 
■720  i;®"'"480  if^'’"l080  1152  ^“*"103680 
and  consequently  the  partial  fractions  are 
86400 1— <j?^"ll520^'^^*^\— 6480^'^^'"^  \—x'^ 2304^"^^^^  1 —x 
43981  / ^ 1 199577  1 
' 103680 a;  “!■  345600  1—x' 
from  which  the  non-circulating  part  is  at  once  obtained. 
7 + 
