300 Remarks on Professor Wallace's Reply to B, 



cess of multiplication of the functions /m, 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 /??i=(l -\-x)"\ 

 yn=(l 4- .t)", therefore their product/m,/n=(l -i-x)"'^". De- 

 veloping this quantity (l-f-x)"'"^", w-j-"? being an integer, by 

 the same formula (A) we shall have when r)z, n, are integer 

 positive numbers, 



m+w m-^nm-{-n~ I m-{-n m-i-n—'i 

 /m,//i=l+— ^-x-}-— J— . ^ .a; = + —- ^ • 



?«-}-« — 2 



Comparing the second member of this expression with liie 

 formula (C) we get the values 



vi-{-7i m-\-n m-\-n — 1 m-f-n m-j-n— 1 m-\-n~^ 



A=-j--,B=-Y- "2 ; C=-— .-— ^— .— — . 



ftic. and by the principle above explained these may be 

 adopted for all values of m and w, so that the formula (D') 

 may be applied to all such values. Now if in the formula (B) 

 we write m-\-n for n we shall get for/(m+'0 an expression 

 equal to the second member of (D') iherefore we shall have 

 for all values of m, n, positive, negative, integer, fractional, 

 or surd, 



fm. fn=f{m-{-n) (E) 



and this is equivalent to Professor Wallace's formula I. page 

 i282, Vol VII. putting ;w=«, n=b. 



Having obtained (he formula (E) Euler easily deduces 

 from it the expressions/^n,/?!, fp=f{m-\-7i-\-j}) ; fm,fn, fp, 

 f()='f{'m-\-n-\-p-\-q) similar to Professor Wallace's /a, fb, fc 

 -/(a+6+c) ; fa,fb.fc.fd=f{a + b+c^d) 



aiso, (fmy=f{ma) similar to his formula H. Vol. VII. page 

 282, 



i i 



/(-)=(! -{-a;)" similar to his formula IV. 



Bn6f(-m) = {l-\-x)"'"for negative exponents, as in page 282. 

 This comparison of the two methods, shows that Euler^ 

 demonstration is identical in principle with that published by 

 Professor Wallace, as B. asserted in his first communication. 

 With respect to the application of the method to the investi- 

 gation of logarithms and exponential quantities, no objection 



