particular  Integrals  in  Cayley's  solution  o/Riccati's Equation.    435 
r    wfl       »-l     1  >-.)(n-3)...(n-2r  +  l)l 
1     ;[r      »-l^r-i"t","V     ;   (n-l)(»-2)...(n— r)ir      / 
J  •       (-lV~       r(n-l)(n-2)...(»-r) 
(„_1). ..(„-,•)  L  (r 
^T  2~^  + 
>i — 1 /n  — 1       ,  \       /?i  — 1 
Cr!-)-(¥^)r.i 
and  the  quantity  in  square  brackets  =  coefficient  of  f  in 
T 
(1  +  *)->-  L_L.  2<(1  +0"-,+  ... 
it- 
=(1+,r.(i-if-^=(i-^)-i 
If  therefore  r  be  odd,  the  coefficient  of  /3r  in  P  is  zero ;  and  if 
r  be  even,  the  coefficient  is 
1  (n-l)(n-3)...(n-r+l) 
r«-l)(n- 
(w-l)(n-2)...(?i-r)  ^r2*r 
1  1 
(n-2)(w-4)...(n-r)  ^2** 
We  therefore  have 
1  /S4 
+  ...,     .  (2) 
n_2"r  (7i-2)(rc-4)222 
1  ff 
(n-2)(n-4)(n-6)  23  3 
which  evidently  satisfies  (1).  Since  then  P  involves  only  even 
powers  of  /3,  and  Q  is  derived  from  P  merely  by  changing  the 
sign  of  /3,  it  follows  that  P  =  Q. 
In  a  similar  manner  it  can  be  shown  that 
R=s=^{1+n^2  +  __l___fl  +  ...}..  (3) 
When,  however,  n  =  2i  +  \,  and  P  and  Q  are  finite  series,  we  do 
not  have  Pt=Q]  (P,  and  Qt  denoting  the  terminated  series  in 
2  F  2 
