456 
cular values of any other function f(a), corresponding to the 
values 1, 2, &c., of x The two methods of developing F (x) 
in a series of factorials, which are here noticed, seem to have ad- 
vantages over the method of indeterminate coefficients, in being 
more simple and direct, and in manifesting more clearly the law 
which the coefficients Ao, A}, Ao, Az) &c., follow. They furnish, 
at the same time, interesting examples of the use of separating 
symbols of operations from their operands ; and it is for this 
latter reason, rather than on account of any novelty in the re- 
sults arrived at, that they are now submitted to the notice of 
Members of the Academy. 
I. Employing u* to denote the pperation which changes 
Fr (x) into F (v+1) we are entitled to write 
F(a +n) =u" F (a) and F(n) = u" F (0). 
But wis known to be equivalent to 1 +A; we may there- 
fore write 
F(n)= (1 + A)"F (0); 
or, with the right-hand member of the equation developed, 
AF Or, sy aoe 
F(n) = F(0) + ——— Pie eset n(n—1) + &e. (1) 
A particular case of this theorem is commonly given in 
treatises on the calculus of finite differences, viz. : 
Ao” 2An 
a = a + TF 0 (el) + &e. 
And indeed the theorem itself may be derived from the 
fundamental expression for w,4, by making « = 0. 
* Arbogast, in his Calcul des Derivations, has appropriated the letter © 
to this use, as being the initial of the word Etat; and in so doing he has been 
followed by recent writers. But against this usage it may be objected that 
the symbol £ is now devoted to a different office in the theory of elliptic func- 
tions. And, on the other hand, there seems to be a peculiar fitness in denot- 
ing by u that operation which changes u, into u 49 
« . 
