14 
[386 
386. 
ON THE LOGARITHMS OF IMAGINARY QUANTITIES. 
[From the Proceedings of the London Mathematical Society, vol. n. (1866—1869), 
pp. 50—54. Read Dec. 12, 1867.] 
The theory of the logarithms of imaginary quantities admits of a remarkably simple 
representation. 
Let P denote at pleasure the imaginary quantity x + iy, or else the point the 
coordinates of which are (x, y); viz., P regarded as a quantity will denote x + iy, 
but we may also speak of the point P. 
Writing thus 
and similarly 
we have of course 
an imaginary quantity X + iY; 
of which are (X, Y). 
P = x +iy, 
P' = x + if, 
P _ x +iy 
P' x + if ’ 
p 
and the point p will be the point the coordinates 
We have 
P = re i6 , 
viz., r is =fx- + y 2 , the radical being positive, and 6 is an arc such that 
a x • /i V 
cos a = — , sin 6 = jl , 
wx^ + y- V a? + y- 
and moreover 6 may be taken to be an arc between the limits —7r, + 7r. The arc 
so defined may be denoted by tan -1 -, so that we have 6 = tan -1 -. 
J J x x