1905-6.] Distribution of the Proper Fractions. 
123 
Hence the fractions */^>w are distributed J(?i + 2) in each of 
3 
the classes except the extremes, which contain n+ — . 
If the fractions 
are counted 
the 
extremes contain only ^(n + 3). 
(2) Into n + 1 classes. 
Introducing the fractions */n+ 1 , the fractions */lf>(n + 1 ) will 
be distributed n + 2) in each class except the extremes, which 
3 
contain n + . The fractions */n + 1 are distributed one in each 
class except the extremes, which contain ~ ; viz., between 
— and -J— we have --- and ( — — ^ and so on. Hence 
n + 1 n + 1 n + 1 \n+V 
removing these we are left with \n in each class except the 
extremes, which contain n. 
If the fractions 
0 1 
1 ’ ’ ' * ’ T ’ 
are counted J, the 
extremes contain, like the other classes, \n. 
(3) Into n + 2 classes. 
Introducing the fractions */n + 1 and */n + 2, all the fractions 
*j^>(n+'2) are distributed h,(n + 3) in each class except the 
extremes, which contain n + -~- . Now the fractions */n + 2 are 
distributed one in each class except the extremes, which contain 
3 . 
-p , and the fractions */n + 1 fall one in each class. Hence 
removing these we are left with \(n- 1) in each class except the 
extremes, which contain n. 
If the fractions 
, . . . are counted J, the 
extremes contain \n. 
Table II., if we strike out the last and the last two columns 
respectively, gives us the distributions of */^>12 and */^>ll 
into 13 classes. 
If we distribute the fractions into more than n + 2, or into 
fewer than n - 1 classes, we shall find that the middle class or 
classes will not have the same number of fractions as the others. 
