298 Remarks on Professor Wallace’s reply to B. 
n ant 4 _ 
frail ypotz Be Si ls SS 37 +, &c.  (B) 
Euler also says that when n is a positive integer number, the 
value of this series is known and is expressed by fn=(1+-2)". 
Its value in other cases he investigates in the following, man- 
ner : 
Changing in (B) the quantity n into m, he gets 
m mm—t1 mm—1lm—2 é 
eee aoe bar gre te n+; Gc. 
] 
Multiplying these values of fn, fm, their product fn. fim will 
evidently be a series ascending according to the integer posi- 
tive powers of w of the following form : 
Sn fn=1+ Av+Br?+Cax?+Da*+Ea5+,&c. (C) 
in which it is plain that the co-eflicients A, B, C, &c., depend 
wholly on m and n, 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 
i in the following manner, in vol. XIX, page 108, Noo. 
‘omm. . 
| fm=1+ 
a 
- _m mm—1 
So ed a 9 +t? +, -ef€. 
‘ n ei 
fr=i+o rato: att, ete. 
{ m mm—ti 
| pete ae x?+, etc. 
| i ae 
4 pety. 7.fet+, ete: (D) 
nn—tl ; 
the more simple expression Jn, which is adopted in this paper, putting 
[n}=fn, [m]=fm, (m-+nj=f (m+n), ete., 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), 2=—kz, n= we obtain the same series 
which Mr, Stainville calls Sa. 
ee 
| 
4 
7 
we | 
Ba 
es 
— 
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