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Proceedings of the Royal Society of Edinburgh. [Sess. 
expected ; hut as there must be some correlation between the two sets of 
results, the great excess is not unexpected. In two cases is between 
1 and 2, and in two between 2 and 3. In two of the latter cases 
the presence of a single individual would make the fit good, and only one 
individual could be expected considering the small numbers observed. 
It may be taken, then, that in these twenty-seven districts at the present 
moment the conditions for the applications of the theory may be held 
to exist. 
In the table just given the value of the ratio ad : be is stated in each 
case. For convenience it will in future be noted by the letter R. It has 
a wide range of variation in value. The lowest value is 4T and the 
highest 233 ; but of the twenty-one different values eighteen lie between 
7 and 11. The mean is 9T4, and the probable error of this ±*48. 
A number, however, such as the ratio at present considered has for each 
individual observation a very high probable error. I have been unable to 
evaluate the expression for the probable error of R in terms of the frequencies, 
and it is difficult to make a reliable estimate of this ; but by an application 
of the formula given by Mr Udny Yule (6) for the probable error of the same 
ratio in the fourfold division, it must be large. The average number of 
observations in each case does not much exceed two hundred, and, taking 
this value and making a rough estimate, it would seem that the probable 
error when R = 9 is 2. That is to say, that in half the cases R should lie 
between 7 and 11. As we have just seen, two-thirds lie in this interval. 
When these ratios are considered from the point of view of the median it 
is found that the latter lies almost exactly in the same place. As small 
values of the ratio are just as likely to arise from emigration as large 
values from immigration, it therefore seems probable that the number 9 
approximately represents the value of the ratio. The only value which 
is possible on the current theory of Mendelism is 7, namely 2 3 -l. The 
observations do not favour this value, so that the latter cannot be taken with 
reasonable probability. 
Leaving Scotland for further verification, it seems best to take only 
large numbers. Dr Beddoe gives eight instances in which the criteria 
demanded in Scotland approximately hold, and in which the numbers 
observed are upwards of four hundred. These are collected in Table V., 
p. 469. 
The mean value of R in the case of these towns is 9 '4, with a probable 
error of ±'67, so that they show no certain difference from the result 
obtained. If anything, they render the value 7 obtained by the 
Mendelians less probable. In the absence of other evidence we may take 
