204 THIRD REPORT 1833. 



general rule for the concurrence of signs. In a similar manner 



we may consider (1)* (a)* as equivalent to («)*; (1)^ (a)^ as 



equivalent to («)^; (1)" «" as equivalent to a" ; (— 1)" («)" as 

 equivalent to (—«)", and similarly in other cases: in all such 

 cases the algehraical quantity into which the equivalent sign 

 or its equivalent factor is multiplied, is supposed to possess its 

 arithmetical value only *. 



The series for (I + x)", when n is a whole number, may be 

 exhibited under a general form, which is independent of the 

 specific value of the index ; for such a series may be continued 

 indefinitely in form, though all its terms after the (« + l)th 

 must become equal to zero. Thus, the series 



(1 + a:)» = (1)" (l + nx + '^^^=^x^ + 



+ ^'^;-^^\--/^'-; + ^\ r- + &c.) 



indefinitely continued, in which n is particular in value (a whole 

 number) though general in form, must be true also, in virtue 

 of the pi'inciple of the permanence of equivalent forms, when 

 n is general in value as well as in formj-. 



This theorem, which, singly considered, is, of all others, the 

 most important in analysis, has been the subject of an almost 

 unlimited variety of demonstrations. Like all other theorems 

 whose consequences present themselves very extensively in 

 algebraical results, it is more or less easy to pass from some 

 one of those consequences to the theorem itself: but all the 

 demonstrations which have been given of it, with the excep- 

 tion of the principle of one given by Euler|, have been con- 

 fined to such values of the index, namely, whole or fractional 

 numbers, whether positive or negative, as made not only 

 the development depend upon definable operations, but like- 

 wise assumed the existence of the series itself, leaving the form 

 of its coefficients alone undetermined. It is evident, however, 

 that if there existed a general form of this series, its form could 



* This separation of the symbolical sign of affection from its arithmetical 

 subject, or rather the expression of the signs of affection explicitly, and not im- 

 plicitly, is frequently important, and affords the only means of explaining many 

 paradoxes (such as the question of the existence of real logarithms of negative 

 numbers), by which the greatest analysts have been more or less embarrassed. 



f If such a series should, for any assigned value of n, have more symbolical 

 values than one, one of them will be the arithmetical value, inasmuch as one 

 symbolical value of 1" is always 1. 



X In the Nov. Comm. Petropol. for 1774. 



