298 Remarks on Professor Wallace's reply to B. 



n n n—\ nn — 1 w — 2 



fn=\-\--x^-—--'X' + ^.-~^.-Y--^'' + , &c. (B) 



Euler also says that when w is a positive integer number, the 

 value of this series is known and is expressed by/n=i(l-f.x)". 

 Its value in other cases he investigates m the following man- 

 ner : 



Changing in (B) the quantity n into m, he gets 



rii m m—\ mm — Im — 2 



Multiplying these values of fn. fin, their product /n, fin will 

 evidently be a series ascending according to the integer posi- 

 tive powers of x of the following form : 



/m./w=l+Aa;+Ba;2-f Ca;3-l-Dx^ + Ea:5-f, &c. (C) 



in which it is plain that the co-efficients A, B, C, he., depend 

 wholly on m and w, and are independent of x. 



This multiplication is precisely like that in Professor Wal- 

 lace's paper at the top of page 279, vol. VII. Euler performs 

 it in the following manner, in vol. XIX, page 108 Kot), 

 Comm. 



^^ m m—l 

 fin=l-{-—.x+-j' -Y-'X^+, etc. 



>• , ■ ^^ *^ n — 1 „ , 



fji=l+j'x-\-- — —' x^-\-, etc. 



r m m m—l 



j l+YVT:+-r--^.a:^+, etc. 



I 



Jm,fn={ r^ + y. 7. ^"^'+5 etc. (D) 



I 



7171—1 



+Y-~^ 'x~-\-, etc. 



Ihe more simple expression /«, which is adopted in this paper, putting 

 [.^]=:>, [m]=fm [m+n]=/(m+n), etc., which is the only alteration 

 made in the symbols used by Euler. It may not be amiss to recall to 



mind that by putting in (B), x=^kz, n=_^, we obtain the same series 

 , wMch Mr. Staiinville calls fa. 



