REPORT ON CERTAIN BRANCHES OF ANALYSIS. 265 



Again, if we consider — a" as originating from (— 1) (+ a) 

 Me shall set 



nt 



\og — a"" = (2 r + 2 7)1 r' + I) Tf \/ — I + m p* : 

 2' 



if we suppose /k = — , r = and r' = — 1, we shall get 



log - x/a =^ p = ^ log « = log i/ a ; 



or the logarithm of a negative quantity will be identical with 

 the logarithm of the same quantity with a positive sign. In a 



similar manner, if we suppose 7)i = i~-~, where p is prime to n, 



r' = — n and r = ^^-^^, then 2 r + 2 7n r' + 1=0, and the 



corresponding logarithm of — a™ will coincide with the arith- 

 metical logarithm of a"'. We should thus obtain possible loga- 

 rithms of negative numbers in those cases in which we should 

 be prepared to expect them from the ordinary definition f of 

 logarithms. 



In the absence of all knowledge of the specific process of de- 

 rivation of quantities, such as a'" and — a"*, we should consider 

 their logarithms as identical with those of l"", A and (—1) 1 *". A, 

 where A is the arithmetical value of a"' : and in considering 

 the different orders of logarithms which correspond to the same 

 value of a"^ or of — «*", they will be found to diflfer from each 

 other by the logarithms of V" and (— I) 1'" only, which are 

 2mr'K V — \ and {2 r -\- 2 m r' -{- 1) * -/ — 1 respectively. The 

 logarithms in qviestion are Napierian logarithms whose base is e. 

 If we should suppose the logarithms to be calculated to any 

 other base, we should replace the Napierian logarithms of 1"* 

 and (— 1) 1"' by the logarithms of those qviantities (or signs) 

 multiplied by the inodiiliis INl : the same remarks will apply to 

 such logarithms which have been made with respect to Na- 

 . pierian logarithms. 



The question of the identity of the logarithms of the same 

 number, whether positive or negative, was agitated between 

 Leibnitz and Bernoulli, between Euler and D'Alembert, and 

 has been frequently resumed in later times. The arguments in 



* Peacock's Algebra, p. 569- 



t The logarithm being defined to be the index of the power of a given base 



which is equal to a given number, it would follow, since ai = + n, that— ;j 



is eijually the logarithm of + n and — h. The same remark aj plies to all in- 

 dices or luyarilhmv which are rational fractions with even dentrainatorb. 



