276 Mr. A. Cayley on the Theory of Logarithms. 



but when x is negative, 



^= tan-'^ +7r: 



X 



TT TT 



where tan~' denotes an arc between the limits — „, +-^, and 



where the upper or under sign is to be employed according as y 

 is positive or negative. I use for convenience the mark = to de- 

 note identity of sign ; we may then write 



6= tan-'^ + eTT, 



X 



where 



x= +, e = 0, 



«■=— , €= +l=.y. 



It should be remarked that 6 has a unique value except in the 

 single case x= — , y = 0, where is indeterminately +7r. We 

 have, in fact, 6= +7r or ^=— tt according as x is considered as 

 the limit oi x + yi, y=. + , or of x-\-yi, y^= —. It is natural 

 to write 



log {x + yi) = log r + 6i, 



or what is the same thing, 



\o^{x + yi)= log '/x^ + y'^+ (tan"' — +e7r h"; 



and I take this equation as the definition of the logarithm of an 

 imaginaiy quantity. The question then arises, to find the value 

 of the expression 



log {x + yi) + log [x' + ifi) — log {x + yi) {x' + y'i) . 



The preceding definition is, in fact, in the case of x positive, 

 that given by M. Cauchy in the Exercices de Mathematique, 

 vol. i. ; and he has there shown that x, x\ xx' — yy' being all of 

 them positive, the above-mentioned expression reduces itself to 

 zero. The general definition is that given in my Memoire sur 

 quelques Formules du Calcul Integral, Liouville, vol. xii. p. 231 ; 

 but I was wrong in asserting that the expression always reduced 

 itself to zero. We have, in fact, in general 



tan~'a+ tan~*/3= tan"' 



when 1— -a/S is positive; but when \ — a.^ is negative (which 

 implies that a, /3 have the same sign), then 



a.-\- B 



tan "'«-)- tan~'/S= tan"' 7, +7r, 



1 — up 



