280 Mr. A. Cayley ow the Theory of Logarithms. 



It thus appears that when the real parts x,x\xx' — yy' are all 

 three of them positive, or any two of them positive and the third 

 negative, E is equal to zero, or the logarithm of the product is 

 e.jual to the sum of the logarithms of the factors ; but that if 

 the real parts ai-e one of them positive and the other two of them 

 negative, then if a certain relation between the real and imagi- 

 nary parts is satisfied, but not otherwise, the property holds; 

 and if the real parts are all three of them negative, the property 

 does not hold in any case. 



The preceding results do not apply to the case where any one 

 of the arguments x + yi, x' + y'i, {x + yi){x' + y'i) is real and ne- 

 gative, for no definition applicable to such case has been given 

 of a logarithm. If, however, we assume as a definition that the 

 logai'ithm of a negative real quantity is equal to the logarithm 

 of the corresponding positive quantity, then it is easy to obtain 

 ^=— , 2/ = 0, 



log X + log {x' + y'i) — log x{x' + y'i) = em, e = + 1 ^ ^ ; 

 an equation which is, in fact, equivalent to 



log {a/ + y'i) — log [ ~ (*'' + y'i)] = em, €= ±l=y'. 

 And xy' + x'y = 0, xx'—yy'=—, which implies y^y', then 

 log {x + yi) -\- log (a?' + y'i) — log {x + iji) [a^ + y'i) = iri, 

 €=±l=y ory'; 

 an equation which is in fact equivalent to 



log {x + yi) + log (— ^ + yi) — log(a?^ -|- y^) = eiri, e = + 1 = y. 



The case where both of the arguments x + yi, x' + y'i are i-eal 

 and negative, i. e. x= — , y=0, jr'= — , ^=0 gives of course 

 \ogx+ log,*?'— \ogxx' = 0, the logarithms of the negative real 

 quantities x, x' being by the definition the same as the logarithms 

 of the corresponding positive quantities. It should, however, be 

 remarked that the definition (^^ — ) log ,2;= log {—x) not only 

 gives for log x a different value from that which would be obtained 

 from the general definition of a logarithm, by considering logo? 

 as the limit of log {x + yi, y^ -\-, or of log {x + yi), y^ —,hu.t 

 gives also a value, which, for the particular case in question, con- 

 tradicts the fundamental equation e^''s^ = x. It is therefore, I 

 think, better not to establish any definition for the logarithm of 

 a negative real quantity x, but to say that such logarithm is 

 absolutely indeterminate and indeterminable, except in the case 

 where, from the nature of the question, x is considered as the 

 limit of x + yi, y positive, or of x + yi, y negative. 



2 Stone Buildings, 

 March 15, 1856. 



