194 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
§ 6 . 
18. Before proceeding further in the search of the fundamental formulae for the 
transformation of operations, we shall exemplify the theory which precedes by invert- 
ing binomial operations and applying the results to some simple cases. 
Let 0, O' denote two linear operations relatively fixed or free, and let us seek the 
value of (0 — 0') _1 - 
Put (0 - 0')- 1 = 0- 1 -j- ni ; the latter being the difference of two linear operations 
must itself be linear. 
Hence 
] = (0- 1 + (0 - 0') = i - 0 - 1 o' + m (0 - &) 
therefore 
Similarly put 
which gives 
whence 
so again put 
m (0 - O') = 0- 1 0 '. 
ill = 0_1 0> 0_1 + 
ni (0 - O') = 0- 1 O' - (0- 1 O'Y + ^ (0 - O') 
y 2 (0 - O') = (0- 1 o') 2 
, 2 =( 0 - l 0')2 0- 1 + , 3 
= (0 _1 O') 3 0~ 1 + 
&c. = &c. 
We thus obtain 
(0 - 0')- 1 = 0~ 1 + (0- 1 O') 0~ l + (0- 1 O') 2 0~ l + + (0 =1 07- 1 0- 1 + % 
where vj n represents the compound operation (0 -1 O'f (0 — O')" 1 . 
The same formula continued to infinity would be obtained by first putting 
O' 1 (1 — 0 -1 0') -1 for (0 — 0') -1 ; and since the operations represented respectively 
by 1 and 0 1 O' are relatively free, we should have by art. 11. 
(1 - 0~ } O')" 1 = 1 + 0- 1 O' + (0 — 1 O') 2 + &c. ad infin. 
When 0, O' are relatively free the theorem becomes 
(0 - 0')- 1 = 0- 1 + 0 -2 o' + o~ 3 o' 2 -f 0~ 4 0' 3 -j-.... o-^o" 1 - 1 + 0- n 0' n (0 - O')" 1 . 
19. For a first example suppose to denote the finite difference, on the supposi- 
tion that by the operation 4^ the quantity x is changed into x + h. 
Then A t ~ j is the finite integral, and in the usual notation of analysts is denoted 
by 2^, we have therefore 
[/»] 2*= [/(*)] (^- I)" 1 = [/(*)] {V 1 + V’ + V s +-4.- M) + V-2,} 
= /(* - +/(* — 2 h) +f(x -3 //) + ... 
+/{« (w — 1) M 4- - nh); 
