STUDIES IN STATISTICAL REPRESENTATION. 483 
From what has been stated we see that as in the con- 
vention, by which so-called “‘imaginary’’ quantities can be 
represented as lying outside the space in which, for “‘real’’ 
quantities, the function in which they arise is representable, 
(e.g., x —1 or xi represented as being at right angles to 
the x axis, or to an «y-plane, etc.) so can the logarithm 
of a negative number arising in an xy—plane be regarded 
as representable, say in the direction of the z-axis, or in 
some other way. This convention remains to be further 
examined. 
Eat 2— —*, then y = 2 => (—)"; and log y =n log’2 
=nlog(—x). We may also put, if necessary, log —% = 
log (—1) + log (+). Similarly, if « = —0, —1,and - 0, 
we may regard the logarithms as numerically equivalent to 
those of +0, +1 and +o but spatially distinguishable 
therefrom by whatever may be implied by log (—1). It is 
possible to represent this spatial distribution by a bifacial 
line or surface, on which as a number passes through the 
values of 0 to —o, the corresponding values of the logar- 
ithms, of the number pass from — © to +0 on one face, 
say the « face (inverted face); and when the number passes 
through the values 0 to +, the logarithms of the number 
pass over the same range in the same manner, but on the 
normal face. Thus log (—1) =+ may be regarded as an 
operator inverting the line or surface on which the quantity 
is representable without numerically affecting it.* In this 
it is analogous toi = }/-1. 
We shall then have log -1 = + log 1; and« (log -1) = 
«log 1 = log 1; or putting a suffix to denote where the 
number of operations is even, viz., p (pair), or odd i (impair), 
log (1) = login nee ely = log) pees (24) 
1 It would be well to retain the Greek letter « to denote this and ana- 
logous operators, and i to denote the operator ,/—14. 
