208 
[107 
197. 
NOTE ON THE THEORY OF LOGARITHMS. 
[From the Philosophical Magazine, vol. xi. (1856), pp. 275—280.] 
An imaginary quantity x + yi may always be expressed in the form 
x + yi = r (cos 6 + i sin 6) = re 91 , 
where r is positive, and 6 is included between the limits — 7r and + tt. 
in fact 
and when x is positive, 
but when x is negative, 
r = V x 2 + y 2 ; 
6 = tan -1 - ; 
x 
6 = tan -1 - + 7r ; 
x 
We have 
where tan -1 denotes an arc between the limits — ^7r, + \tt , and where the upper or 
under sign is to be employed according as y is positive or negative. I use for con 
venience the mark = to denote identity of sign; we may then write 
where 
6 = tan -1 - + err, 
x 
x = +, e = 0, 
X = — , € = ± 1 = y. 
It should be remarked that 6 has a unique value except 
x = —, y = 0, where 6 is indeterminately + 7r. We have, in fact, 
in the single case 
0 = + T Ol’ 9 = — 7T