386] 
ON THE LOGARITHMS OF IMAGINARY QUANTITIES. 
17 
value of the right-hand side is thus = log p. As regards the left-hand side, the 
indefinite integral is in like manner = log;s; but here, if the chord P'P cuts the 
negative part of the axis of x, there is a discontinuity in the value of log z, viz., 
if the chord P'P, considered as drawn from P' to P, cuts the negative part of the 
axis of x upwards, there is an abrupt change in the value of log z from — iir to + iir; 
and similarly, if the chord cut the negative part of the axis of x downwards, there 
is an abrupt change from + iir to — iir; in the former case, by taking the definite 
integral to be log P — log P', we take its value too large by Pnr, in the latter case 
we take it too small by 2iir; that is, the true value of the definite integral is in the 
former case = log P - log P' - 2iir, in the latter case it is = log P - log P' + 2iir. But 
if the chord PP does not cut the negative part of the axis of x, then there is not 
any discontinuity, and the true value of the definite integral is = logP-logP'. We 
have thus in the three cases respectively 
p 
log P - log P' = log p + 2iir, 
= \ogp-2i7r, 
= lo gp> 
which agrees with the previous results. 
It may be remarked, that it is merely in consequence of the particular definition 
adopted that there is in the value of log P a discontinuity at the passage over the 
negative part of the axis of x; with a different definition of the logarithm, there 
would be a discontinuity at the passage over some other line from the origin; but a 
discontinuity somewhere there must be. For if, as above, the chord P'P meet the 
negative part of the axis of x, then forming a closed quadrilateral by joining by right 
P P 
lines the points 1 to P, P to P', P' to p, and ^ to 1; the only side meeting the 
c ^ 
negative part of the axis of x is the side P'P; the integral j —, taken through the 
closed circuit in question, or say the integral 
has, by what precedes, a value in consequence of the discontinuity in passing from 
P' to P; viz., this is = — 2iir or = 2iir, according as the chord P'P, considered as 
drawn from P' to P, cuts the negative part of the axis of x upwards or downwards; 
but this value — 2i7r or + 2iir must be altogether independent of the definition of 
the logarithm ; whereas if, by any alteration in the definition, the discontinuity could 
be avoided, the value of the integral, instead of being as above, would be = 0. The 
foregoing value — 2iir or + 2iir is in fact that of the integral taken along (in the one 
C. VI. 
3