204 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
Represent this product by 
+ A x rf*- 1 + A 2 ^- 2 + . . . + A m x n ~ m + . . . + A n _ 1 x + A n , 
the general coefficient A m being the sums of products, each of which contain m factors. 
For A l it is easily seen that its value is 
n + ( n — 1) a 2 + (n — 2) a 3 + . . . 2 a n _ 1 + u H . 
Again, A 2 consists of products such as a 1 a 2 , a l a 3 , a 2 a 3 , &c., and pure powers, as 
eq 2 , a 2 2 , &c. ; the general form of the first class of terms is a p a , and we now proceed 
to find its coefficient, or the number of times this combination occurs, which number 
may be denoted by (x p a ), and supposing p less than q, no factor preceding x -f- eq 
+ a 2 + . . . will be concerned in forming the combination in question, and in the 
factor itself and the succeeding ones the terms preceding a may also be neglected. 
The factors commencing from the above, arranged horizontally, will form this 
diagram. 
x + + • • • + <* p 
x + a i + • ♦ • + a p + a p + 1 
X + a l + • • • % + Up + 1 + % + 2 
0C + «i + . • • «p + + l + «p + 2 + • • • + % • * 
^ + «i + • • • «p + + + a ? + 1 
x + + • • • % + a q -j- a ? + i + a q + 2 , 
&c. &c. 
Now if a ?+1 were placed where the asterisk stands, the combination of a p with a q and 
a + i would be alike, .*. (a p a g ) — (a p a ?+1 ) = the number of combinations of one 
term at the asterisk with the terms in the vertical column of a p , except that a p which 
is the same horizontal line with the asterisk ; it is therefore the number of terms 
minus one in that column which (since p — 1 factors precede the first above written) 
will be n — p. 
Therefore A denoting the finite difference, when q increases by unity we have 
A («p %) — -(n-p); 
therefore 
{a p a ) = (n — p) (c — q), c being independent of q. 
Suppose q — n, (a p u n ) will be the number of terms minus one in the column of 
since a n enters only once; that is (a p a n ) — n — p, therefore c — n — 1, or c = n -f 1 
which gives 
( a p %) = ( n — p) (n— q + 1). 
As for the coefficients of the powers as cc p 2 , denoting such by a similar notation (a p 2 ), 
they will not be affected by the supposition that 
oq = 0 a 2 = 0 . . . u p _ x = 0 oi p + j = 0 . . . = 0, 
