1899] METHOD OF TREATING VARIATIONS 415 
From this theoretical conclusion we are led on to the practical out- 
come. If the conditions are altered we get a second group of 
individuals each of which conforms to the same law, but the centre of 
the averages is altered. Hence if any individual be chosen at random 
from one of these groups we should be able to tell by an examination 
of its characters to which group it belonged. Hence the following 
practical rule. 
3. The sum of the squares of the variations in the characters of 
a certain group is a minimum for the individuals of that group. 
This follows directly from the equation of the probability integral, 
1 — 2x2 
ie 
the nearer y approaches the centre of the curve, or the average, the 
smaller z? becomes. This being true for all the characters, we have 
that =z is a minimum for the variations in the characters of a certain 
group. Hence, if it is desired to know to which of several known 
groups a certain individual belongs, it is necessary to calculate the 
variations of each character of the individual from the respective 
averages of the several groups, then find the sum of the squares of 
these variations, and the least sum shows the group to which the 
individual is most nearly allied. The more characters that are taken 
the more likely is the result to be right, but less characters are neces- 
sary the greater the number of. individuals. The first example taken 
to illustrate this is one of several given by Prof. Heincke. It refers to 
a single specimen of the group of herring obtained from the White 
Sea, which had 58 vertebrae where the average was 53°6. One might 
think, therefore, that this individual was abnormal for this group, or 
belonged to quite another group. ‘Two other groups are therefore 
taken, the one from the west and south-west coast of Norway (Vaarsild), 
which has 57°5 as the average number of vertebrae, the other from 
the Jutland Bank off Denmark, which has 56°6 vertebrae on the 
average. When other characters are considered, however, and the 
variations of this single individual form the averages of the three 
groups calculated according to the method, we find that 
From the average of the White Sea (35 characters) . . (a or)v? =3°213 
i Vaarsild (35 characters . : p v2 = 3°696 
5 * White Sea (37 characters) . : v* = 3°225 
¥ = Jutland Bank (37 characters) : Mists I 
In each case the least value shows that this individual more closely 
approaches to the herring of the White Sea in spite of its having a 
seemingly abnormal number of vertebrae. From this it follows that 
whilst in one character an individual may be very much above the 
average, it has a variation or group of variations in other characters 
below the average, which balance by “defect” what the first has in 
“¢ excess.” 
