STUDIES IN STATISTICAL REPRESENTATION. 481 
Again, if 
DG SALE OS AME 4 Cait. a (20) 
Ra A AC) = ASIC) (AH YAY); ete: 
=y = GeAxX”" = Gem” = CA” = CAM oo... (21) 
where log p = Y'; and log € = C, log A = A; etc. 
This last curve may be called the second anti-logarithmice 
generatrix of y = a + mx, and if € be unity, the first anti- 
logarithmic generatrix of equation (20). 
Provided its axes of reference are suitably determined, 
a curve therefore is the anti-logarithmic generatrix of its 
logarithmic homologue, and the logarithmic homologue of 
its anti-logarithmic generatrix. 
It is important to observe that the logarithmic homologue 
depends not only upon the form of the curve, but also wpon 
its position with relation to the axes of reference. 
Similarly, if ("CAO ee ene OSI (22) 
its anti-logarithmic generatrix is 
a ee (23) 
where, therefore, log Y = y: and so on. 
8. Logarithms of negative numbers.—Since log 0, log-+1, 
log + © are respectively —«, 0and +o, the whole range 
of negative and positive real numbers is exhausted in 
expressing the logarithm of the numbers +0 to +0. 
Moreover, with a positive number as base, no power, 
positive or negative, integral or fractional, can give a 
negative number. It is consequently usual to say that in 
general there can be no logarithm of a negative number. 
Any curve which may be represented by negative or 
positive numbers may therefore have an anti-logarithmic 
generatrix. We proceed to consider whether every curve 
can be said to have a logarithmic homologue. 
Again, in order to follow out the matter a little more 
closely, consider the expressions 
y = x"; and log y = » log %, or 7 = né. 
Ex—December 2, 1914. 
