376  Mr.  R.  F.  Grwyther  on  the  Range  of 
If  we  write  out  in  full  the  resulting  expression  for  F(  — ^  V 
the  successive  terms  will  be  seen  to  show  a  marked  resem- 
blance to  those  in  the  expansion  of 
{l-Me~)~\ 
In  order  to  examine  this  resemblance  more  closely,  I  expand 
J  F(— ~\  >       on  the  assumption  of  convergence,  and  find 
e    c 
{F(^)}-3  =  l-3/i;"-(3/^f/1a) 
+(i*'-i*7  '  +(hi~lhJ  c  +  im 
+  TAV* (4) 
It  is  on  the  greatly  increased  convergency  produced  in  the 
series  by  this  step  that  the  argument  depends.  It  is  clear 
that  there  is  a  value  of  a,  not  very  widely  different  from  3A. 
for  which  l—fjuec    is  a  factor  of  the  right-hand  side  of  (4). 
For  more  precision,  write 
3A  =  u/jl  +  /3/j?  +  y/JL* 
and  determine  a,  /3,  y  so  that  the  right-hand  side  of  (4)  may 
ik<p 
vanish  approximately  when  1/fi  is  put  in  place  of  e  c  ?  i.e.  so 
as  to  satisfy 
29 
!_?* J-+ =0. 
This  leads  to  the  equations 
K1-Aa2-^3)=-(^a4+iia5+iAi^) 
