MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
193 
more generally 
Secondly, since 
therefore 
0 4 -* = 4 ~ n . 0 4 -»• 
X ' X i X XiX 
0A=A04+0A 
X X XXIX 1 X X 
A - 1 0 T A r — 0~fl + A 0~A 7 
X XX XIX 6 X XI 
Put now 0 X 4* -1 for 0 X , and therefore 0 X for 0 x $ x ; hence 
therefore 
A -1 0 4 -1 A = 0 + A - 1 . 0 4 _1 A 
X X TX X X 1 X X T X X 
0 A - 1 — A - 1 .0 4 - 1 — A -1 . 0 -A - 1 . A . A -1 . 
X X X X ‘X X X T X X X 
Put 6 1 f° r ® x -> thence we have 
0 A - 1 — A - 1 .0 4 _1 — A ~ 2 .0 4 “ 2 A . + A “ 2 . 0 \L ~ 2 A 2 . A 
V x X x X ' X X X T X X • X X < X X X J 
or we continue this process indefinitely 
0 A ~ 1 = A ~ 1 . 0 4 _1 — A “ 2 . 0 4 “ 2 A 4- A ~ 3 .6 \L ~ s A 2 —■ See 
u x x x x tx x x t x x 6 x x t x x 
which is the same as the general formula for 0 x A x when n — — 1. 
Again 
but 
and 
A ~ 2 = A ~ x . 0 - 1 . A — A _1 . 0 A - 1 A . A “ 2 
X X XIX X X XIX XX* 
A- 1 0 4 - 1 . A _1 = A “ 2 . 0 4 “ 2 — A “ 2 . 0 4 - 2 A . A - 1 , 
X X T x X X XIX X X IX X X 1 
Hence 
A “ 1 . 0 4 - 1 A . A _1 = A “ 2 . 0 4 - 2 A — A “ 2 . 0 4 ~ 2 A 2 . A - 1 . 
■X X TX X X X X T x X X X T X X X 
0 A “ 2 = A ~ 2 . 0 4 “ 2 — 2 A “ 2 . 0 4 “ 2 A . A -1 + A 2 . 0 4 “ 2 A 2 . A 
x x x x Tx x x t x x x 1 x X T X “ar a 
and in a similar manner it is easy to prove generally in a terminating series 
.! n{n— 1) 
0 A A ~ n .0 4 ~ n - n A ~ n ,9 4 “ B A . A 4- -\-~.0 4 A 2 . A ~ 2 - 
X 1-1 X V X T X X X T X X X ' 1.21 x ' x X X 
or in an infinite series, 
0 A ~ n — A ~ n .0 4 -"-wA -( B + 1) .04 “"( n + 1 >.A -p n ^'V > 1 ~A -(*+ 2 )04”-(’ , + 2 >A 2 . 
X “ X X X T x X X T X X ' 1.2 x XTX ^ X 
Thirdly, divide A x by h , and then put h — 0, whence 
0 x d~ l = d~ l . 0 x — . 0^ d x d~ l 
-l a ,1 -2 7i 7 i .7 -3 
~d-\0~ d~ 2 .0X4* F . 0 r d 2 - &c. 
a? a? a? x x 6 x x x 
9 d~ n = . 0 - » d ~ n . FJ . d ~ * -J- • M? • ' 2 - &c. 
a? a? x x x xxx , 3.2 x x x x 
= . 0, - » £~ (w+1) Ml + • i -(n+2) - FF 2 ~ &c. 
a? a? a? j n > j 9 2 x x x 
which formulae admit of most extensive applications, whether 9 X be regarded 
quantity or a fixed operation. 
-2 
9 
- &c. 
-&c. 
as a 
