particular  Integrals  in  Cayley's  solution  ofRiccafcV&Equation,    437 
and  write  /S2  for  a  finally  after  the  differentiation ;  then 
fd  Vr  _/d  V  f    _,  af  a'+*  1 
UJ  v/a~U«^a  ffi+1-    (2i+3      "'J 
=A0a-*-*  +  A2^-\ ..  +A2**"1— B0+A2i+2a*— B2«  +  .. *, 
in  which  it  is  to  be  remarked  that  the  terms  in  a,  a2, . .  .  a*-1 
give  no  terms  on  differentiation.  There  are,  as  it  were,  two 
series  whose  coefficients  are  distinguished  by  the  letters  A  and 
B,  the  latter  commencing  from  the  term  a ,  and  including  oti+1, 
«,+2,  &c.     AYe  thus  find 
P1=A'0  +  A'2/32  +  A'4/3*+  . . .  -/8"(B'0  +  B'tfl*+B'4|8«+  . . . ). 
and  it  will  be  shown  that  the  first  series  =  P,  and  the  second 
=  aR.     To  prove  this,  if  r<i, 
2  jr 
so  that 
A,    _,_  y _1 ,  _I_. 
A»~H     )  (n^2)(n--4)...(n-2r)  2r(r* 
if  r  be  greater  than  i,  then 
2_i 
A2r=1.3...(2r-W)2^ 
and 
A'2r=(-)1  S(l.3...(n-2)\{l.3...(2r-n)\y'[r; 
so  that  the  first  series  =  P,  as  is  evident  from  comparison 
with  (2).     The  second  series 
=     n-w      JLji+ *±I & 
1  .3.5...(ra-2)   '[2i+l  I         (n  +  l)(n  +  2) 
(t  +  2)(i+l)      /34  , 
+  (7H-I)...(7i  +  4)C2"t"-,,| 
~  (J  .  3  .  5  . . .  tzJ2  I        «  +  2    2  ^  (7i  +  2)(7i  +  4)22(2^-"J 
=  «R;  therefore  PX  =  P  — «R; 
and  by  merely  changing  the  sign  of  /3  we  have  Q^Q-f  «R, 
whence  (5)  follows  at  once.  If  n=  —  (2i-f  1),  it  will  be  found 
that 
R  =  Ili  +  «P,     S  =  S,-«Q; 
so  that 
