1905 - 6 .] Studies in Immunity : Theory of an Epidemic. 485 
Later, he applied the same method to the case of smallpox in 
1871-2, and fitted a curve to the latter portion of the epidemic. 
His description of his method is not clear, hut in a paper by Dr 
G. H. Evans {Trans. Epid. Soc., 1874-5) it is given in detail. 
The method practically amounts to assuming that the second 
difference of the logarithms of successive ordinates of an epidemic 
curve is a constant, and using a value of this constant, deduced from 
an early portion of the epidemic, to predict the succeeding portion. 
The method in the terminology of finite differences is as 
follows : — 
If u = log y where y is the ordinate of the epidemic curve, then 
A 2 u — - c (a constant by Dr Farr’s supposition) 
of which the integral is 
M= -^l+Az + B 
or as log y = u 
cx 2 a ™ 
— o 4" Ex + B 
?/ = e 
which is the equation to the normal curve of probability. 
Dr Farr does not seem to have noticed that the application of 
his arithmetical law leads to this curve. As a matter of fact, it is 
a very good approximation to the middle parts of some epidemics, 
though it does not provide a specially good fit for the whole 
course of those to which he applied it. In the examples of this 
method given by Dr Evans nothing more is attempted. The real 
difficulty in the application consists in finding a good value of the 
constant from the early portion of the epidemic. 
This is all the literature of this subject. My attention was 
specially drawn to this matter when considering recently some 
questions of immunity. The interpretation of some of the facts 
required accurate knowledge of the epidemic processes. I was 
struck, when I began to examine the course of epidemics, by the 
close resemblance which many bore to the probability distributions 
developed by Professor Pearson ; and, without any knowledge that 
Dr Farr had already come indirectly to fit the epidemic curve to 
that of the normal frequency of error, I applied the methods now 
in use of fitting probability distributions to statistics. 
