804 
MR. J. W. L. GLAISHER ON RICCATI’S 
and, writing the operators in a different order, this expression 
_™-2 i ™2?' ( L ™-2z'+l 'r—'r ~1 r \ 
- 1 Aj _ tX/ • tty 7 tty • i/y _ tty • • • tty , tty • tty I tty J 
dx dx dx dx ' 
/ d \ 2i+1 
=^ + \£) <M4 
This investigation of (25) is in effect the same as that given in the last article, but 
the form in which the process is presented is somewhat preferable. 
Denoting for the moment the operator x l+n j-x~” by [n~\, then 
[a][a+6] . . . —l)5]<£(.r)=.r 1+ "— x n .x 1+a+b ~x " h . . x} +a+{l 1)b yx n {l 1)b .(f>(x) 
dx 
dx 
- ytO Idi 1 . 
d V (p(x) 
d x xa +(i-1)f 
Also, writing the operators in the reverse order, 
[a+({—!)&] 
Thus 
[a-J-6] [a]<f)(x)=x i+a+{i 1)5 Jbc a {l l)i . . . x 1+n+l ^x a b .x 1+a ^x a .<f)(x) 
■ y-tO ?l/1 xyt ^T* 1_ 
d y (f>(x) . 
dx x a 
r a-b( r b+l d \_m_ a+ib( T -b + \ d V 
x 1 dxh? + «-W- r dx X- 5 
and therefore, replacing by </>(,r), 
or, writing iff- 1 for i, 
(--l) <J>{x)=x (i+ w(x b+l £j x {i 1)b (f>{x) 
y + 1 p) (^[x)=x (,+2Vj (x * + y) y^(x). 
• (27), 
(28), 
34. Putting l>=— 2, and <f>(x) = y and e" r respectively in (27) and (28), these 
formulae give 
0 dV c a ' r 1 / 
