1905 - 6 .] Distribution of the Proper Fractions. 119 
the sum of which is A; 2 * J r\r'- 1), and there are here k integers, 
hence 
[± 1 + • I — 1) 1^ 
L r J * L r J r * r kr 2 
n 
r 
1/ , x 1,, n \ , lX 1 
-r{r+\)-—hr = - . - r(r + 1) - — r 
as before : similarly if n is not prime to r+ 1. 
Hence in each class except the extremes there are \{n + 1) fractions, 
and in the first class there are the n fractions 
0 0 
® and 1, 
n n 
1 ’ 2 5 ' * 
which is counted J, i.e. altogether n + J. 
We shall refer to this distribution of the fractions into n 
classes as the normal distribution. 
If the series of improper fractions, positive and negative, be 
joined to the ends of the series of proper fractions, then the 
fractions -5-, . . . ; -i-, must each be counted only J, 
and the number of fractions in each class without exception is the 
same, i.e. 1). 
§ 2. We shall now develop some theorems relating to the way in 
which the fractions */p are distributed throughout the classes. 
In any class there can occur , besides the limits , no two fractions 
For — , 
P 
~s' = l. 
— =£ £ which is greater than — unless p = n and 
p p n 
If in any class */p and */n — p both occur , they must both be 
equal to one of the limits of the class. 
We have 
r .. s 
— -f* - 
n p 
1 
and 
l > _±_ > r±L 
n n — p n 
Therefore and — > r + 1 ~ l ' 
r n r + 1 r ‘ n r + 1 
