300 Remarks on Professor Wallace’s Reply to B. 
cess of multiplication of the functions fm, fn, to obtain the 
series (D) is the same whatever m and n may be, whether 
integral or fractional. Now in the case of m and n being 
integer positive numbers, the formula (A) gives fm=(1+2)", 
Jn=(1+<)”, therefore their product fm, frn=(14-x)""". De- 
veloping this quantity (1-+-«)"*", m-+-n, being an integer, by 
the same formula (A) we shall have when m, n, are integer 
positive numbers, 
mtn mtnm+n-1 
. 2 ° 
fis, fist + 1 @+ 
feta? | 
3 
~~ m+n m-+-+n-i 
a? . 3 —e 
‘x3 +&c. (D’) 
Comparing the second: member of this expression with the’ 
formula (C) we get the values 
m+n, m+n m+n—1_ m+n m-+-n-l m+n-2 
Be hee A ei eh ge 
* , 
etc. and by the principle above explained these may be 
adopted for al/ values of mand n, so that the formula (D’) 
may be appliedto all such values. Now ifin the formula (B) 
we write m-+-n for x we shall get for f (m-+-n) an expression 
equal to the second member of (D’) therefore we shall have 
for all values of m, n, positive, negative, integer, fractional, 
or surd, 
fin. fr=f(m+n) (KE) 
and this is equivalent to Professor Wallace’s formula l. page 
282, Vol. VII. putting m=a, n=6. 
Having obtained the formula (E) Euler easily deduces 
from it the expressions fim, fn, fo=f(m-+n-+p) ; fm, fn, fp, 
/q=f(m+n+p+q) similar to Professor Wallace’s fa, fb, fc 
=f(atb+e); fa, fb, fe, fd=fla+b+e+d) 
lag (fm)*=/(ma) similar to his formula II. Vol. VIE. page 
Sp ? 
z é 
A= +a) similar to his formula IV. 
and f(-m)=(1+-2)-"for negative exponents, as in page 282, 
this comparison ef the two methods, shows that Euler’s 
demonstration is identical in principle with that published by 
Professor Wallace, as B. asserted in his first communication. 
ith respect to the application of the method to the investi- 
gation of logarithms and exponential quantities, no objection 
