197] 
NOTE ON THE THEORY OF LOGARITHMS. 
209 
according as x is considered as the limit of x + yi, y = +, or of x + yi, y = ~. It 
is natural to write 
log (x + yi) = log r + di, 
or what is the same thing, 
and I take this equation as the definition of the logarithm of an imaginary quantity. 
The question then arises, to find the value of the expression 
log (x + yi) 4- log (x + y'i) — 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 Exercises de Mathématique, vol. I. [1826]; and he has there shown that 
x, x', xcd -yy' being all of them positive, the above-mentioned expression reduces itself 
to zero. The general definition is that given in my Mémoire sur quelques Formules 
du Calcul Intégral, Liouville, vol. xn. [1847], p. 231 [4.9]; but I was wrong in asserting 
that the expression always reduced itself to zero. We have, in fact, in general 
when 1 — a/3 is positive; but when 1 — a/3 is negative (which implies that a, ft have 
the same sign), then 
where the upper or under sign is to be employed according as a and ft are positive 
or negative; or what is the same thing, 
where 
1 - a/3 = +, e = 0, 
1 — aft = —, e=±l = a + ft = a = ft. 
This being premised, then writing 
C. III. 
27