MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
205 
< n ___ '7)J (tI ~~~ m r p | | j 
they are therefore the same as in ( 1 + "“' + 1 that is which 
is 
half of the formula obtained by putting q — p. 
Hence 
Ao = 
n.(n — 1) 
. a. 
+ 
(■ n — 1) (n — 2) 
9 , ( n - 2 ) ( n ~ 3 ) 9 , 
• a 2 H T2 * a 3 + 
1.2 1 ' 1.2 
+ (w — 1 ) (n — 2) a x a 3 + (n — 1 ) (w — 3) a 4 a 4 -f- . 
-f- (n — 2) (n — 2) a 2 a 3 -j- (n — 2) (n — 3) a 2 a 4 + . 
+ (n — 3) (n — 3) a 3 a 4 + . 
. {n — 1 ) (w — 1 ) a 4 a 2 
In like manner we may classify the terms of which A 3 is composed into terms of 
the forms a p a q a r , ee , a ; ; 3 respectively, p, q, r being arranged according to magni- 
tude ; their coefficients may be represented as before by the same letters in brackets. 
Every combination of a a q may be combined with a r , except such as are formed 
from the a p and u q which are in the same horizontal line with it, if these are erased 
the number n is reduced to n — 1, and the combinations of a p a q are then by what 
has been already shown only {n — p — 1) (n — q) in number, therefore the excess of 
the number of the combinations of a r with ct p a q above that of a r + 1 is (n—p—l) ( n — q ) 
or taking the finite difference in reference to r 
A K % a r) — - (n-p - 1) (n - q) ; 
thence 
“q^r) — (n—p — \) (n - q) (c - r), 
and putting r — n we find as before c = n + 1 ; therefore 
{«p %“r) = (n — p— 1) (n—q) (n - r + 1) 
and generally if s > r, r > q, q > p, &c., then by the same process 
(u s a r a q a p ) = (n — s + 1) (n — r) (n — q- 1) (n—p — 2) 
Again, if we erase the a p which is in the same horizontal line with a r the number of com- 
binations of the remaining terms a p (in number n — p) are — — — - and since 
the number of terms in the vertical line where a stands is n — q + 1, it follows that 
(«p i .2 • ( n q ~ J - 1)> 
and generally 
(//a 9 ...) 
\ s t q / 
_ ( n — s + 1) (n — s) ... (s 1 times) (n — r) (n — r — I) .. . (r 1 times) (n — q — 1) (ti — q — Q) ... g’ times 
1 . 2 ., 
1 .2...H 
1 . 2 . . . q 1 
2 E 2 
