438  On  Cayley's  solution  of  Riccati's  Equation. 
a  being,  as  before,  only  suitably  modified  on  account  of  n  being 
negative. 
The  connexion  between  the  integrals,  if  we  regard  the  series  as 
terminating  when  the  numerators  of  the  factors  become  zero, 
is  therefore :— (1)  if  q  is  not  =  ±  (2i  +  l)~\,  then  P  =  Q,  R  =  S, 
and  P  and  R  are  independent ;  (2)  if  q=  (2i  + 1)-1,  P  and  Q  are 
independent,  and  E  =  S=J"- (Q-P);    (3)   if  q=-(2i  +  l)-\ 
R  and  S  are  independent,  and  P=Q=—  (S— R).     But  the 
proper  way  is  to  regard  the  series  as  including  all  their  terms ; 
and  then  the  result  merely  is  that  we  have  always  two  indepen- 
dent particular  integrals  P  and  R,  such  that  P  +  «R  and  R  +  dP 
are  finite  algebraical  expressions  in  the  respective  cases  of 
q  =  (2ifl)-1  and  ^=-(2i  +  l)-1. 
The  integral  made  use  of  in  the  { Quarterly  Journal '  to  ob- 
tain the  above  expression  for  P2  was  due  to  Poisson,  and,  altered 
so  as  to  agree  with  the  notation  of  (1), takes  the  form 
r°°  _  2    x2q 
u=  \    e         w^dz, 
which  by  a  pair  of  simple  transformations  (which  are  given  in 
the  paper  referred  to)  may  be  written  in  either  of  the  forms 
I        vn-le-v2-iPr-2dv>    or    £»l       Vn-l 
Jo  Jo 
-\e-\^~v^dv. 
By  expanding  the  exponential  factor  in  the  former  of  these  and 
integrating  each  term  by  means  of  the  integral 
j%*-*«irp+!),  ....   (6) 
we  obtain  the  series  (2).     A  similar  treatment  of  the  second  in- 
tegral and  reduction  by  means  of 
j: 
fM^e-»-v«)=ir(— \^) 
(7) 
leads  to  the  other  series  (3).  Of  course  this  method  is  .not  le- 
gitimate, as  in  both  cases  we  must  at  length,  whatever  q  may 
be,  come  to  some  point  where  the  factor  that  multiplies  the  ex- 
ponential (in  the  form  e~v2)  under  the  integral  sign  is  raised  to 
a  power  <—  1,  when  the  term  becomes  really  infinite  and  the 
method  fails.  But  it  is  remarkable  that  if  we  ignore  this  failure 
and  treat  (6)  and  (7)  as  if  they  were  universally  true,  we  are  led 
to  both  the  series  (2)  and  (3),  and  thence  from  a  particular  to  a 
general  integral  of  (1). 
Trinity  College,  Cambridge, 
May  12,  1872. 
