REPORT ON CERTAIN BRANCHES OF ANALYSIS. 205 



be detected for any value of the index whatever, which was 

 general in form, and therefore, also, when that index was a 

 whole number ; a case in which the interpretation of the opera- 

 tion designated by the index, as well as the performance of the 

 operation itself, was the most simple and immediate. 



That such a series, likewise, would satisfy the only sym- 

 bolical conditions which the general principles of indices ini- 

 poses upon the binomial, might be very easily shown; for if 

 m and n be whole numbers, then if the two series 



{\ \ xf -V^\\ ^■ mx ■\- -\ ^ x^ -j- &c. J 



(1 + a-)« = 1« (l + n X + '--^^^-^^^2 +- &c.) 



be multiphed together, according to the rule for that purpose, 

 we must obtain 



(1 4- xy^''^ l™-*-" ( 1 + (m+ n)x + ^ \2 ^^7 



a series in which m + n has replaced m or n in its component 

 factors : and in as much as we must obtain the same symbolical 

 result of tliis multiplication, whatever be the specific values of 

 m and n, it follows, that if the same form of these series repre- 

 sents the development of (1 -|- or)" and (1 + x^, whatever m and 

 n may be, then, likewise, the series for the product of (1 + a:)"' 

 and (I + xY, or (1 + x)'"+", would be that which arose from 

 putting wi + w in the place of m or n in each of the component 

 factors. If, therefore, we assumed S {uri) and S (ra) to represent 

 the series for (1 + xj" and (1 + x^, when m and n are any 

 quantities whatsoever, then (1 + a-)" x (1 + .r)« = (1 + ^)'« + " 

 = S (w -f- w) = S (;h) X S (w) ; or, in other words, the series 

 will possess precisely the same symbolical properties with the 

 binomial to which they are required to be equivalent. 



It is the equation a"" x «" = a*""*"", for all values of ra and n, 

 which determines the interpretation of «"' or a", when such an 

 interpretation is possible ; in other words, such quantities pos- 

 sess no properties which are independent of that equation. The 

 same remark of course extends to (1 + xj", for all values of w, 

 and similarly, likewise, to those series which are equivalent to 

 it. That all such series must possess the same form would be 

 evident from considering that the symbolical properties of 

 (1-1- J")" undergo no change for a change in the value of n, and 

 that no series could be permanently equivalent to it whose form 



