466 Proceedings of the Royal Society of Edinburgh. [Sess. 
This sum is denoted by the symbol y 2 5 the value of P or the probability 
that the fit might be worse is then obtained from the published tables (5). 
For the above figures P = '67; that is, if 1400 persons were observed by 
random sampling 100 times, in 67 of these a worse fit might be expected 
than that found. In the case of the upper line the fit is practically 
perfect. 
In what follows, the figures relating to Scotland are chiefly used. Con- 
cerning the suitability of these it may be remarked that (excluding Glasgow 
and Edinburgh, where the recent Irish immigrations have introduced a, 
large element unassimilable on account of the difference in religion, and 
which therefore fulfil none of the conditions necessary to the application 
of the present theory), Dr Beddoe made observations in 43 localities in 
which the characteristics of the hair and eyes were noted in more than 
150 persons. 
If 43 cases are noted at random, the number of good fits and bad fits 
may easily be calculated from the probability table already referred to.. 
We find y 2 should be less than unity in ’393 of the cases; greater than 
unity and less than two in '239 ; greater than two and less than three in 
T25; and greater than three in the remainder, namely, *223. The 
following table is divided into two classes — the towns with the larger 
districts, and the country districts. It is seen that the number expected not 
only is realised but largely exceeded ; in other words, except for the fact 
that the number of towns and large districts in which y 2 is greater than 
three is twice that expected, the number of small values of y 2 is much in 
excess of that required. The exception is to be expected as into these 
towns specially the immigration has been much the greatest in recent 
years. 
Table III., showing the Distribution of the Forty-three Districts 
in Scotland according to the Actual Findings and the 
Theoretical Proportions expected by the Theory of Chance. 
Values of x 2 - 
0-1. 
1-2. 
2-3. 
3-. 
Towns and ( A ctual . 
6 
2 
5 
large districts \ Theoretical 
5T 
32 
1-9 
2-9 
Small districts ( ^ tual ' , ' 
1 theoretical 
22 
11*7 
3 
7T 
2 
4*4 
3 
6-7 
T f i / Actual . 
iocai| Theoretical 
28 
5 
2 
8 
11*7 
10-3 
63 
9*7 
