50 
ME.  A.  CAYLEY  ON  THE  PAETITION  OE  NOIBEES. 
these  are 
where 
§ — X § — X 
XiS 
X2i 
=coeff.  7 in  7 ^_7s 
t f[qe  0 
= coeff.  7 in 
t fiqe  *) 
and 
1 
— A_2  7^+^-i  7+  &C., 
Avhere  it  is  easy  to  see  that 
A -J- 1 
A. 
and  we  have 
^2?=— f‘'(l-f)A_2 
But  [1— gi^]  = l+g'4-g>^=0.  Hence  f*=l,  and  therefore 
(1  -rf(i  -fxi  -?‘)(i  -f“)= (1  -fKi  -f')== 9. 
Hence 
whence 
;=  - Tf2f’(i -f) = A2  (1 -?’)= A2  (2+?)- 
^2^ — 
and  the  partial  fraction  is 
which  is 
1 2 +x 
162  l+x  + x"^' 
1 2 — X — x‘^ 
and  gives  rise  to  the  prime  circulator  jgo  (2,  —1,  —1)  per  3^. 
The  reduction  of  is  somewhat  less  simple ; we  have 
2g^  4§  5§‘^ 
\4  18Vl-g~l— §2  ' 
— r;|^4  18  1-5  18  1-5^ 
