182 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
tion A of finite differences and that of dividing by h being both of that character, 
the compound A . y is generally linear, and must remain so in the limiting state 
when h vanishes. Hence every function of a linear operation is itself of the same 
class of operations. 
5. The composition of polynomial linear operations is effected in the same manner 
as algebraic multiplication ; with, however, this peculiarity — the order of the com- 
pound operations, when not relatively free, must be strictly preserved. 
Thus let 0, O' i, / represent linear operations : then 
M (0 + 0') (/ + |') = [u] (Oi + O' , + 0,' + OH) ; 
for / / being linear will act on each of the parts [w] 0, [u] O', which form their subject. 
Again, when 0, O' are relatively fixed, [w] (0 O') 2 will not, as in algebraic involu- 
tion, be identical with [u] (0 2 + 2 0 O' + O' 2 ), its correct value being [u] ( 0 2 + O' 6 
-f- 0 0' + 0' 2 ) ; which, however, is the same as the former expression, when 0, O' are 
relatively free ; for then O' 0 = 0 O'. 
Similarly 
(0 + O') 3 = (0 3 + O' 0 2 + 0 O' 0 + O' 2 0) + ( 0 2 O' + 0' 0 0' + 0 O' 2 + O' 3 ) ; 
or putting 0 ( "' ) O' for the sum of the compound operations in which 0 twice enters, and 
O' and 0 ,( " ) for the sum of those containing 0 once and O' twice, this may also be 
written 
(0 + o') 3 = 0 3 + 0 (2) 0' + 0 0 ,(2) + O ' 3 ; 
and employing a similar notation, we shall have, when n is a positive integer, 
(0 + 0') n = 0 {n) + 0 (n_1) O' + 0 (n ~ 2) 0 ,(2) + 0 (2) 0 l(n ~ 2) + 0 0' {n ~ }) + 0 ' (n) . 
The term 0 (,i_1 ^ O' in this formula is the sum of n terms, formed by placing O' at the 
beginning, at the end, and in all the n — 2 intermediate positions of the expression 
0 0 0 (n — 1) times. Similarly, by the known theory of permutations, 0 (u-2 ' 1 0 ,(2; 
is the sum of ■■■ „ — - terms, &c. 
Hence when 0, O' are relatively free, we have 
(0 + 0') n — 0 n + n 0 n ~ l O' 4- U ■ y - ~ 7 - ) . 0”~ 2 O' 2 4- .... n 0 0'”" 1 4- 0' n . 
6. The following theorems are immediately deducible from the general expansion 
of the preceding article : 
Since A = — 1, therefore 
A"=(4 
. \n in i n — 1 . 71 [ll — 1 ) . n — 2 
1 ) — n \J/ 4 - - \ 2 .4 
n (n— 1 ) ( n - 
1 . 2.3 
•+(-!)" 
or, if we introduce the subject f (x), and observe that [/ (4)] 4- n h), Ihen 
A n f{x) =f{x + nh) — nf {x + (n— 1 ).h} 4- -/( T + (n—2) A}— ,&e. (I.) 
