16 
ON THE LOGARITHMS OE IMAGINARY QUANTITIES. 
[386 
and similarly 
log P' = log r + id', 
and 
Hence 
\ogP-\ogP' = \og~ + i(0-0'-(f)), 
so that, by what precedes, log P — log P', if the chord P'P, considered as drawn from 
P' to P, cuts the negative part of the axis of x upwards, is = log p, + 2iir; if the 
P 
chord cuts the negative part of the axis of x downwards, it is = log p, — 2irr, and 
P 
in every other case it is = log -p. 
P' to P, cuts the negative part of the axis of x upwards, is = log ™ + 2iir ; if the 
It is to be remarked that log P, as above defined, is a continuous function of 
P (= x + iy), with the single exception that, if the point P move from above to below 
or from below to above the negative part of the axis of x, the imaginary part of 
the logarithm changes from + itr to — iir, or from — iir to + iir, in the two cases 
respectively. And we are thus led to another mode of looking at the question. 
Consider the integral 
The value of the integral may depend on the series of values assumed by the variable 
z as it passes from the limit z — P’ to the limit z = P, or say it may depend on the 
path of the variable z; in order to give the notation a precise signification, we must 
therefore fix the path of the variable z; and I do this by taking the path to be 
the right line P'P. Write now z = P'.u, we have — = - ; z = P’ gives u = 1; z = P 
z u ° 
p 
gives u = jp; and it is easy to see that, the path of ^ being along the right line 
; and it is easy to see that, the path of ^ being along the right line 
P' to P, that of u is along the right line 1 
the point the 
coordinates whereof are # = 1, y = 0, to the point p-, 
We have thus 
f p dz __ l" p + p/ du 
Jp> z~ Ji u 
the path in each case being a right line as above. The indefinite integral 
P . ..... 
and as u passes from 1 to p, there is no discontinuity m the value < 
and as u passes from 1 to there is no discontinuity in the value of log u; the