MATHEMATICAL SECTION. 
85 
therefore serve as a measure of their efficiency. The general problem 
may be stated as follows: Having given the number of instances 
respectively in which things are both thus and so, in which they 
are thus but not so, in which they are so but not thus, and in which 
they are neither thus nor so, it is required to eliminate the general 
quantitative relativity inhering in the mere thingness of the things, 
and to determine the special quantitative relativity subsisting be¬ 
tween the thusness and the soness of the things. 
If no special causes are in operation, the number of instances in 
cth 
which both A and B occur should be = — according to the general 
s & » 
thedry of probabilities. The operation of such causes is therefore 
indicated by a difference between the values of c and —• Now, 
instead of the ratio -, which varies directly as c, let us form a 
ratio which shall vary directly as c — ~ and which shall — 1 
when c = 
These conditions give us the ratio 
ah 
ah 
s 
or 
cs 
ah 
a (s — h) 
In like manner, instead of the ratio 7 we have , , 
b h (.5 
cs — ah 
a) 
low, as¬ 
suming that whenever either of these derivative ratios is constant, 
the required discriminate association ratio varies directly as the 
other ratio varies, and that it equals unity when they each equal 
(OQ _ 
unity, it is found equal to their product = —^ 
For the phenomena A and Not-B the number of concurrences 
probable from general causes = subtracting this quan¬ 
tity from a — c, from a, and from s — 6, the indiscriminate 
association ratio - ^77 becomes ■ , which differs 
a{s — b) ah {s — a) {s — b) 
from the discriminate association ratio for A and B merely in the 
algebraic sign of the quantity under the exponent. This is as it 
should be. Sight is precisely as favorable or unfavorable to hear- 
