1905 - 6 .] Distribution of the Proper Fractions. 
117 
the various fractions, each loaded with its respective weight, is as 
nearly as possible the same. 
The following paper is an attempt to find the evenest distribu- 
tion in any case that may arise. 
I. 
§ 1. Consider all the proper fractions whose denominators do 
not exceed n. For shortness we shall denote any fraction with 
denominator p by */p, and by the assemblage of all the 
proper fractions whose denominators do not exceed n. 
. 0 1 0 12 
The whole number of fractions i.e. -p -pi ->y> 
is equal to 2 + 3 + . . . + (n + 1) = ^n(n + 3). 
If the fractions n are distributed into n classes , 0/n to 1/n, 
1/n to 2/n, . . ., (n— l)/n to n/n, and any fraction which falls 
between two classes is counted h in each of these two classes , each of 
the others being counted 1 in the class in which it occurs , then in 
each of the classes there will fall + 1) fractions, except in the 
extremes , ivhich contain n + 
Take any class, say that between — and ^ . If — is a 
n n p 
fraction belonging to this class, 
r 
n 
> — > 
p 
r+ 1 
n ’ 
or 
< 
ns 
r + 1 ’ 
To each value of s there will correspond a certain number of values 
of p. Since p cannot be greater than n, it is evident that s cannot 
be greater than r+ 1, and if s — r+\, p must = w, giving the 
r _i_ i 
fraction , which is counted 1. 
n 
Let — denote the greatest integer less than — ; if — is an 
L r J r r 
integer, there will be a fraction — = — which is counted A : in 
& p n 2 
this case we shall define f— j = — - A . 
