ON THE LOGARITHMS OF IMAGINARY QUANTITIES. 
15 
386] 
It is to be observed that 9 has always a determinate unique value, except in 
the single case y = 0, x negative, where we have indeterminately 6 = ±ir. 
It is further to be remarked that, taking A for the origin of coordinates, we have 
9 = angle xAP, considered as positive or as negative according as P lies above or 
below the axis of x. 
Starting from the equation 
we have similarly 
and 
P = re ie , 
P' = r'e id \ 
P 
P' 
where </> is derived from 
P 
F 
in the same way as 9 from P, or 9' from P'. 
Consequently 
and therefore 9 — 9' — (f> a multiple of 27t, say 
0 — 0' — cf) = 2m7r, 
and in this equation the value of in is determined by the limiting conditions above 
imposed on the values of 9, 0', </>. To see how this is, suppose in the first instance 
that the finite line or chord P'P, considered as drawn from P' to P, cuts the negative 
part of the axis of x upwards; P is then above, P' below, the axis of x; that is, 
9, — 0' are each positive; and drawing the figure, it at once appears that the sum 
9 + (— 0'), that is 9 — 0', is a positive quantity greater than nr. And in this case the 
angle </> will be equal to 2tt — (0 — 0') taken negatively, that is, </> = — {27r — (0 — 0')}, 
or 9 — 0' — cf) = 2-7T. But, in like manner, if P'P cut the negative part of the axis of 
x downwards, P will be below, F above, the axis of x; —9 and 9' are here each 
positive, and the figure shows that the sum — 9 -F 9' is greater than it ; and in this 
case the angle </> is = 2ir — (— 0 + 0'); that is, we have 0 — 0' — <f> = — 2?r. In every 
other case, (that is, if the chord P'P either does not meet the axis of x, or if it 
meets the positive part of the axis of x,) 9—0' and cf) are each in absolute magnitude 
less than 7r, and we have 0 — 0' — <£ = 0. So that we see that, according as the chord 
P'P, considered as drawn from F to P, meets the negative part of the axis of x 
upwards or downwards, or as it does not meet the negative part of the axis of x, 
the value of 9 — 0' — </> is = 271-, = — 2tt, or = 0. 
Taking now log r to represent the real logarithm of the positive real quantity r, 
we may, as a definition of the logarithm of the imaginary quantity P(—x + iy), write 
log P = log r + id. 
The value so defined is of course one out of the infinite number of values of the 
logarithm, and it may for distinction be termed the “ selected ” value. In all that 
follows, the symbol “log” is to be understood to denote the selected value. We have 
log P = log r + i9,