190 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
also since p (x) = p (x + h) change x into x + h p (x J r h) = p (x 2 h) and 
mnn V — 1 
generally p (x) = p (x + m h). m being an integer; and since p (a;) = Cs ml 
satisfies this equation, it follows that + &c. satisfy the equation (a.) ; 
the complete appendage will therefore be 
nx */ — 1 2 n x */ — 1 — me — 1 — 2 « V — 1 
Aj S A ~b A 2 2 A 13 1 £ * “b 2 A -|- 
the number of constants being infinite. 
§5. 
15. When the simple operations which compose a compound one are relatively 
free their places are transmutable, but when fixed a mutation of places will require 
an alteration in the operations themselves. 
Let denote the changing of x into x + h as before, and let 6 x be any operation 
affecting x, or, which is the same, fixed relati ve to 4> x > then considering 6 r as a sub- 
ject, put \0~\ \L V = 0[ v , and if the compound operation [u] 0 x \p x be proposed, its value 
by transmutation is \ii\ \p x 6' x , for in the first compound operation ^ x affects all the 
preceding symbols as forming its subject. 
Again, let jV] 6 X A x = y be proposed for transmutation, we have 
y - |>] 6 X (A r - 1) 
= M Wr 
and putting A r + 1 for ^ x , and 6 X A x for the finite difference of 0 x considered as one 
operation, we have 
y = [u] (A t & x + 0 x A x ) = [w] 6 X A x . 
Lastly, divide this identity by h , and then put h = 0. When -j becomes d x , and d’ x 
becomes & x , we get for the transmutation of 6 X cl x , 
M K d r — M ( d v K + K d J- 
These formulae of transmutation separated from the subject are respectively 
0 -A — \L/ 0 \L/ 
X T x T X X ~ X 
0 A = A <nJT 4- 0~A 
XX X X I X ' X X 
0 cl — d 0 “j“ @ d ’ 
X X XX' XX 
When 0 X is constant or not dependent on x, then 
TF = 0 Fa = o T~d — o, 
and these formulae will then express merely that 0 x , \J/ x , &c. are relatively free. 
For example, let 0 X simply denote a multiplier, then 0 x \p will be 0 x + k , also a mul- 
tiplier, 0 x A x — 0 x + h — 0 X , which may be represented by '0 X and 0 X d x = the limit of 
i±L — when h vanishes, or the differential coefficient of 0 X , which may be denoted 
by 0' x , then the general formula becomes 
