530 FOURTH REPORT — 1834. 



stance, that though f'^ a + f~ ' a ■=/' ' (a a), yet, as f~ ' a 

 + f~ ' (I, in its indefinite form, admits the addition of any one 

 value to any other value o^f~ ' a, it has twice as many values as 

 2f- ' a : that /- ' (a^) =2 i -k + 2/" ' «, and that/- ' 1 +/- ' 1 , 

 or generally f~ ' 1 +/" ' a, considered as an indefinite formula, 

 is precisely equivalent to/" ' 1 or /"-' a respectively. 



(f = 1'* cfg or generally a = l'^«!f , «. e. all the values oidf 



are given by multiplying any single value in succession by all 

 the values of 1'. Now V has an infinite number of values, 

 unless j; be a "rational fraction" {positive or negative, inclu- 

 ding integers,) in which case the number of values is equal to 

 the denominator of the fraction in its lowest terms. 



If a' have among its values two quantities differing only in 

 sign, X is a rational fraction, with, in its lowest tei'ms, an even 

 denominator. Let a be positive and x a rational fraction, 



which in its lowest terms = — , the number of real values of 



n 



a" will be one or two, according as 7i is odd or even. Let 

 ci' = y, then x, if a be negative and y positive, must be a ra- 

 tional fraction, with, in its lowest terms, an even numerator 

 and odd denominator ; if a be positive and y negative, an odd 

 numerator and even denominator ; if a and y be both nega- 

 tive, an odd numerator and odd denominator. When x is of 

 the form r + V —\s, a real, and r irrational, a* can have 

 only one real value. When a is real, r rational, and s not 



= 0, a*^ ^ ~^ *, if it have one veal value, has an infinite num- 

 ber. When X is of the form \/ —\ s and a real, whenever 

 one value of a'" is real, all the other values, of which there are 

 an infinite number, are also real. 



A quantity {p + \^ —\ q) may have no real logarithm, and 

 can have no more than one in a given b ase (r + >»/ — \ s), un- 

 less the " moduli" of the quantity (= i/p* + q^, adopting the 

 phraseology of M. Cauchy,) and of the bass are both = 1, in 

 which case the number of real logarithms is infinite. When 



one real logarithm exists, and one only, it is = y~^ '^ — %• 



When an exponent is real and rational, and in such case only, 

 it will reappear at intervals with different ranks in different 

 orders, as a logarithm of the same quantity in a given base. 



In conclusion, the author states, that, as all the values of 1* 

 were before known (at least when x was real) to be comprised 

 in the formula cos {2i x-n) + \/ — I sin {2 i x ti), the principal 



