124 
Proceedings of Royal Society of Edinburgh. [sess. 
§ 5. In the normal distribution and in the three just considered, 
when we reckon the fractions at the extremes as 1, the number 
in the extreme classes is practically double that in the others ; 
but when we count these fractions as \ the number in all the 
classes is the same both for the normal distribution and for the 
distribution into n + 1 classes, and for the other two distributions 
the number in the extremes is J more than in the other classes. 
On the former convention, we may, if n is even, in the normal 
distribution make pairs of classes coalesce, starting with the 
second ; then there will he n + 1 in each of the \(n - 2) compound 
classes and n 4 - J in the extremes. 
If n is odd, we may divide into n + 1 classes, and then make 
pairs of classes coalesce; then in each of the \(n — 1) compound 
classes and in each of the extremes there will be n fractions. 
These are therefore the evenest distributions on the assumption 
that all the fractions have equal weights. 
§ 6. Let us now give weights to the different denominators, or 
consider the denominators not as occurring with equal frequency 
but with relative frequencies denoted by g p . We have now to 
€ p = 1 or J, we have to find the distribution which makes this 
sum approximately the same in each class. It will be easier to 
proceed in the reverse way and find for any distribution what 
relations must hold between the coefficients g p in order that the 
sum in each class should he the same. We shall assume in what 
follows that for the fractions at the extremes e = J. (Without 
this assumption the results require considerable modification for 
the extremes.) 
Let us consider first the normal distribution. 
Taking any two classes, we have the equation 
II. 
reckon each of the fractions *jy not as 1, but as g p , and, if it is 
the limit of a class, as \g p . Then taking the sum ^ e p/V w ^ ere 
i i 
[fj. n will not enter into this equation, since e n = e n = 1.] 
