developement of exponential functions, &c. 
Q 9 
that is, 
— . . x x - l - 
1 m 2 • • • X I« 2 • ^ M I 
whence it is plain that 
and of course that 
X 
(*- 
-0 J 
I) • 
I 
I 
1 . 2 ... 
• •y 
f- <J- 
-0 
1 . 2 ... 
+ &c. = l ; 
K = — 
x,y 
a y o x 
" I - 2. ? * 
where A-^o* denotes the first term of the y th differences of 
the terms of a series o*, i*, z* , &c. We have then making 
tea. 
**/(■*) 
dt x 
D/( i) 
Ao*-j" 
D*/(0 
I. 2 .... X 
A* o x . 
If we separate the symbols of operation from those of quan- 
tity, the second member of this equation may be much more 
elegantly written as follows : 
!“+^ + &}/(>).»■;■■- (O 
referring the D to the functional characteristic/*, and the A 
to the o and its powers.— Or, we may throw it into the fol- 
lowing form, 
|m i ) A + D^ i) A * + ^ai> A q 0 * 
Upon this, we have to observe — first, that the addition of the 
term/ (l) at the beginning of the series within the brackets 
makes no difference in the result ; adding only to it the term 
/(i)xo , which vanishes of itself: and, in the next place, 
that we are at liberty to suppose the series continued to 
infinity ; as every term beyond y A * ° vanishes, owing 
