436       Mr.  J.  W.  L.  Glaisher  on  the  Relations  between  the 
this  case),  as  is  evident  from  the  solutions  of  particular  equations 
written  down  by  Professor  Cayley  in  his  paper. 
The  relation  in  this  case  can  be  inferred  thus  :  the  series  which 
multiplies  the  factor  in  P  consists  of  a  finite  portion  including 
the  first  i+1  terms;  after  this  point  the  terms  vanish,  owing  to 
the  presence  of  the  factor  n— 2i—  1  in  the  numerator,  until  the 
same  factor  makes  its  appearance  in  the  denominator,  when  the 
two  factors  cancel  one  another  and  the  series  recommences ;  this 
latter  portion  is 
M)(n-3)...2(-2)...(-n  +  l)  ffl»      n  +  l      +l  1 
(*-l)(n-2)...l  l(^      (»  +  l/*      ^'"^ 
which  after  reduction  becomes 
(-yr2 ? ^(i+yifl+MM)?+iii}i(4) 
{     }      (1.3.5...rc)2P  1    +w+lP+ (/i+l)^4-2)  (2 +,,,J-  w 
Denoting  then  by  a  the  constant  factor  of  this  series,  (4)  =  «R, 
as  the  series  is  the  same  as  that  in  R,  and  we  have  P  =  Pj  +  «R. 
Similarly  Q=Q1— aS;  and  since  R=S  (these  series  not  termi- 
nating in  this  case  of  w  =  2*  +  l),  we  have,  since  P  =  Q,  when  the 
whole  series  are  taken  into  account, 
P1+GR=Q1-flR, 
and  therefore 
B=Sj(Qi-Pi)  =  S (5) 
The  equality  of  P  and  Q  depending  merely  on  an  algebraical 
multiplication  (each  being  equal  to  (2)),  we  have  good  reason 
for  asserting  that  they  are  equal  so  long  as  all  the  terms  are 
taken  into  account.  It  is  desirable,  however,  to  prove  the  truth 
of  (5)  by  the  direct  development  of  P,  and  Qx  in  powers  of /3; 
and  this  can  be  effected  by  means  of  a  theorem  which  I  proved 
in  the  'Quarterly  Journal  of  Mathematics '*,  vol.  xi. p.  267, and 
which  in  the  notation  of  this  paper  may  be  stated, 
p  =       (-^       (\  AY  c* 
1      1.3.5...(/i-2)V/3  dfiJ    |3 
To  develope  the  continued  differential,  let  /32=a,  so  that 
\p  dp)  (3    ~r  \dJ  v/« 
*  "On  Riccati's  Equation."  The  value  in  the  text  for  P  is  derived 
from  the  equation  numbered  (12)  on  p.  271  by  making  the  same  substitu- 
tions  as  those  used  in  this  paper,  viz.  /3=  —  and^  =  w-1,  so  that  x=finn-n 
and  o?22-i^=w-i/3c?/3. 
