REPORT ON CERTAIN BRANCHES OF ANALYSIS. 265. 



Again, if we consider - a^ as originating from (-!)(+ «)% 

 we shall get 



loa _ a'» = (2'r + 2 w r' + 1) IT >/ — 1 + "' P = 



if we suppose .« = 1, r = and r' = - 1, we shall get 



1 1 



or the logarithm of a negative quantity will be identical with 

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

 similar manner, if we suppose m = j^, where 2> is prime to «. 

 .' = - « and r = ^^, then 2 r + 2 ,» / + 1 - 0, and the 

 corresponding logarithm of - «- will coincide with the arith- 

 metTcariocarrthin of a-. We should thus ohi^m posszble \og^- 

 Stlms of negative numbers in those cases m which we should 

 be I'epared^to expect them from the ordinary defimtionf of 



^T'^rabsence of all knowledge of the «Peeificpvoeess of de- 

 rivation of quantities, such as «- and " f f^ \^ ^^^^°^^^ l^.^^^^l" 

 their logarithms as identical with those of 1-. A and (- 1)1 .A, 

 where A is the arithmetical value of f : and in considermg 

 ThedifFerentorders oflogarithms which cori^^^^^^^^ 

 value of a^ or of - «'", they will be found to diftei tiom eacli 

 otrby the logarithms o? 1'" and (- JLl- only which are 

 2 „, , ^ '^:n and (2 r + 2 m / + 1) tt -/ - 1 respectively. The 

 Togarithms in question are Napierian logarithms whose base i s . 

 If^e should suppose the logarithms *« ^e calculated to any 

 other base, we should replace the Napierian logarithms of 1 

 and r- n '"by the logarithms of those quantities (or signs) 

 multUed by the modulus M: the same remarks will apply to 

 ruKSrithms which have been made with respect to Na- 



■^"itiSw the identity of the logarithms of the same 

 nmnber, whether positive or negative, was agitated between 

 Le bnitz and BernouUi, between Euler and D'Alembert, and 

 has been Sequently resumed in later times. The arguments m 



; ?r;i:Ja?i£^b';i'nrdSed to^e the index of the power of a given base 

 which is equal to a given number, it would follow, since ai = ± «, that— 

 is ecmally the logarithm of + n and - n. The same remark a] plies to all in- 

 dicTor Uan7/L which are rational fractions with even denumnato... 



