206 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
Lastly (a^ 3 ) is the same as if all the terms oq a 2 &c. were zero, except a p and is 
therefore 
(n —p 4- I ) (n — p) (n — p — 1 ) 
1.2.3 
(n — p + 1 ) (n — p) ... (p 1 times) 
More generally 
( a p ) 1.2 . . .p 1 
We have thus investigated the coefficients of every combination which enter the 
d P b , 
whole product, and then if only l be substituted for any general symbol {cxP), the 
dxP P 
required development is completely obtained. 
It may be remarked that the coefficients of the combinations of consecutive terms 
are pure powers, thus 
(cq a 2 ) = (n — l) 2 a 2 a 3 = (n — 2) 2 , &c. oq a 2 a 3 = (n — 2) 3 . 
29. By the preceding investigation we have obtained the following general formula 
in which the subject is any function of x : 
v i • v 2 • v 3 • • • v , x x v J v 2 d, x 1 v ' v 3 a x 1 v n a x j 
+ rf/- 2 {(n- 1)(«- i 
,dx 
d v x d v 4 
+ («- 1) (— »)& & + 
. / ^ dvn dv o , , d v q dv 4 
+ («- 2 )(«-V*'^ + ( "' 2)(, " 3) M''vS + 
+ ( w „ 3) 0 — 3) — ^ + ... 
+ 
+ 
n (w— 1 ) d 2 iq (w — 1) {n — 2) c? a v 2 
2 ‘ v l dx‘ 
+ 
2 
* v 2 dx 2 
(n — 2)(n-3) 1 
2 * v 3 dx 2 * ' ‘ ’ j 
+ 
rf ”- 3 { (« - a) (- - 2) (« - 2) • +. to. j 
+, &C. 
Pat v 4 = v 2 = v 3 = . . . = v n = v hence, 
4 d ) n = d n v + d n ~ l v n ~ l . ^ 
\ x/ x ' 2 * dx 
+ 
v n - 2 J (”- l ) n ( n + 1 )(3”- g ) - 2 ^ , 
^ \ 2.3.4 dx 2 "i" 
(n — 1) . n (« + 1) 
2.3 
„ i d 2 vl 
‘ v dx*\ 
