Prof. Guthrie on >/ — 1. 283 



The next extension of the meaning of the word power, which 

 it is necessary for our present purpose to consider, is that by 

 which fractional indices are explained. According to the defini- 

 tion adopted for this purpose, every power may be considered as 



p. 

 having for its index a numerical ratio as x q ; and such a power is 

 defined as being that ratio multiplication by which q times is 

 equivalent to multiplication p times by x, the need for a separate 

 definition of the word root being thereby dispensed with. 



It is on account of this extension that an inconsistency first 



p 

 arises in the theory of exponents. Theoretically x q should be 



2p p 



an exact equivalent for x 2q ; whereas if q be an odd number, x q 

 has but one sign, whereas x 2q is ambiguous ; so that it might be 



2p 



that the true answer to a question might appear in the form x 2q , 



p 

 whereas the form x q would be incorrect. Such an inconvenience 

 can only be guarded against by a consideration of each particular 

 case, contrary to the principle which should be arrived at in ma- 

 thematical processes generally. 



The next step is to give a meaning to the expression x n when 

 the index n is any ratio whatever, commensurable or incommen- 

 surable. This gives rise to the conception of the exponential 

 function, of which, however, there may be several definitions. 

 One definition is, that by x n where n is any ratio we mean the 

 function which satisfies certain functional equations, as that 

 <£>(#, n) X(f>(x, m) = <f)(x J m + n) &c. The other is, that by x n 

 where n is incommensurable, we mean the limit of x m where m 

 is a commensurable ratio, when m approaches to equality with n. 

 The former or functional definition is the one we shall adopt. In 

 either case we are compelled, for the sake of writing, to consider 

 all powers as having ambiguous signs, so that x 1 =+x. This 

 is not always made apparent ; but that it really is the case may 

 be seen from the allied logarithmic function, where we are com- 

 pelled to consider log ]0 ( — 10) as well as log 10 10 as being equal 

 to unity. According to the exponential definition of a power, 

 therefore, ( — l)^ is by no means an impossible quantity, since 

 ( — 1)^ and ( -f 1)* are both of them + 1 . As, however, it would 

 be inconvenient entirely to dispense with the ordinary algebraical 

 definition of a power on account of its connexion with the process 

 of multiplication, we have to use two different definitions of 

 powers at the same time, one of which includes the other. To 

 avoid the misunderstanding which might possibly arise, let us 

 for the present distinguish these functions as follows. Let x 2 



