192 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
This general formula may be more readily deduced by considering that p x is the 
operation of changing x into x n h, and consequently p x may at once be substi- 
tuted in the first formula for p x . 
Secondly, 
therefore 
but 
0 A — A 0 p -1- 0 A 
XX X X • X 1 X 
x^ 
0A 2 =zA0P A + 0 A A 
X X X X Tx X 1 X X . 
' X ’ 
0 ip A — A . 0 p 2 + ^ P A 
X ‘X X XX *X 11 or T # 
X * 
writing ^ p x for ^ ; and similarly 
whence 
0 A A = A . 0 p A 0 A 2 , 
XXX X X ‘ X X [ X X } 
generally suppose 
0 A 2 ~ A 2 . 6 r P 2 + 2 A . 0 p A + 0 A 2 . 
xx xx ix * a: j? * a: a? ■ xx 
0 A n ~ l ~ A 71 - 1 .0 P 71-1 + (n— 1) A n ~ 2 .0 P w " 2 A 
X X X X T X 1 v / J? v x Tx X 
+ 
(w-l)(»-g) 
1 .2 
. 0 p n ~ 3 A 2 . 
X T X X 
Now if we write 0 X p x for 0 x in the fundamental formula, we have 
0 p m A = A . 0 p m 4- 0 p m ~ 1 A 
X Tx X X X TX 1 X T X x 
each term when we put for m, n — 1, n — 2, &c. successively, will thus be divided 
into two, which being placed in two distinct lines will give 
K K = a;.?a , + (*-i)a;- 1 - w 1 A, + .A /- 2 -w* a} + &c. 
+ A/- 1 . w 1 A, + ( -^M . A/- 2 . 
= vw+»a;-' -w 1 a, + '-44 • a,”- 2 • 
which is the general formula sought for. 
Divide now by h n and observe that y becomes d x and p x becomes unity as a mul- 
tiplier when h = 0, hence the third general formula 
e x d x = d x °x + n d x~ i * e x d x “5“ 'T72” ' d x~~* • 6 x d x + &c - 
which when 6 x represents quantity is the theorem commonly called Leibnitz’s. 
17. We next proceed to investigate the formulae for negative indices. First since 
p~ l denotes simply the changing x into x — h, we may write p ~ x for p x in the first 
formula. 
d x ^x 1 = 1 • °x 1 
Therefore 
