456 



cular values of any other function /(x), corresponding to the 

 values 1, 2, &c., of x. The two methods of developing f (x) 

 in a series of factorials, vehich 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 Aq, Ai, a^, A3, &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 operation which changes 

 F (x) into F (ic+l) we are entitled to write 



F (a; + «) = M" F (x) and f (n) = m" f (0). 



But u is known to be equivalent to 1 4- A ; we may there- 

 fore write 



F («)=(! + A)''f(o); 



or, with the right-hand member of the equation developed, 



F{n) = E(o)-^^I^n^^^n(n-\) + &c. (1) 



A particular case of this theorem is commonly given in 

 treatises on the calculus of finite differences, viz. : 



Ao" A^o" , 



a;" = -— X -\- —— X {x— 1) -j- &c. 



1 I •^ 



And indeed the theorem itself may be derived from the 

 fundamental expression for M^+m by making x = 0. 



* Arbogast, in his Calcul des Derivations, has appropriated the letter e 

 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 e 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 



