434       Mr.  J.  W.  L.  Glaisher  on  the  Relations  between  the 
q-l  (g-l)(8g-l)  1     - 
1+2^T)*,+  2(?_1)2?(2?-1)*     +"f    " 
«=P  =    | 
1      ?(?-J)       gte-i^^-i)  J 
1        5(2  +  1)  9(5  +  l)2?(25  +  l)  J 
If  g(2«  +  l)  =  l,  P  and  Q  are  finite  series,  and  the  complete 
integral  is  w=CP4-C'Q,  a  finite  expression.  Similarly,  if 
<?(2z  +  l)  =  —  1,  w=CR-{-C'S,  a  finite  expression.  This  solu- 
tion therefore  not  only  points  out  the  integrable  cases,  but  also 
exhibits  the  integral  itself,  and  on  this  account  is  preferable  to 
any  other  that  has  been  given. 
Now  suppose  q  is  unrestricted,  we  have  four  particular  inte- 
grals, P,  Q,  R,  S ;  and  since,  if  u= A  and  w  =  B  are  two  indepen- 
dent particular  integrals,  the  complete  integral  is  w  =  CA  +  C;B, 
it  follows  that  only  two  are  independent,  and  that  the  other  two 
are  linear  functions  of  them.  It  is  the  general  determination 
(viz.  (1)  when  q  is  not  =  ±(2i+i)~\  (2)  when  q=  (21  +  1)"1, 
and  (3)  when#=  ~(2i4-l)-1)  of  the  independence  of,  and  rela- 
tions between  P,  Q,  R,  S  which  it  is  the  object  of  this  note  to 
ascertain. 
1  x^ 
It  will  be  convenient  to  write  n  for  -  and  8  for  —  :  the  series 
q  q 
thus  become 
L        n  —  1         (w— l)(n—Z)[2  J 
q-J-!      n-l         (n-l)(n-8)fl»  "1 
R-fn   "  +  lfl,   (n  +  l)(n  +  8)^  "I      „ 
S_/T      n+l  s.(n+l)(n+3) /?  1 
I        n  +  \P+  (n  +  l)(n  +  2)[2       '"f    ' 
If  »  be  not  of  the  form  ±(2i  +  l),  it  will  now  be  shown  that 
P=Q  and  R  =  S. 
The  coefficient  of  /3''  in  P  is 
