1905 — 6 .] Distribution of the Proper Fractions. 
121 
§ . — 
fraction will lie in the (Q s -t- l)th class, and the fraction */p 
will not lie in this class ; but if F s = 0, lies h in the Q s th 
n -p 
and J in the (Q s + l)th class, and — also lies \ in each of these 
classes, where s' = Q s - s. 
This gives a simple way of writing down the normal dis- 
tribution. It is convenient to write the classes horizontally, and 
the fractions *jp in columns under their respective denominators. 
The numerators s in any column must then occur consecutively 
from 1 up to p. We note also that the distribution is symmetrical 
about the middle horizontal line, so that, the upper half having 
been filled in, the lower half can be written down. In filling- 
in the right-hand half containing the larger denominators it is 
•easiest to determine the vacancies by dividing n, 2 n, ... by 
n-p. The other half can then be filled in in a complementary 
way ; or we may proceed vice versa, filling in the left-hand side 
first. The fractions which limit two classes may be distinguished 
by a bar. Thus the normal distributions for n= 12 and for n— 13 
are represented thus : — 
I. 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
0 
1 
1 
1 
1 
1 
1 
1 
T 
2~ 
1 
1 
1 
2 
2 
2 
2 
2T 
3" 
I 
1 
2 
2 
2 
3 
3 
3 
3 
4 
1 
2 
2 
3 
3 
4 
4 
4"_ 
I 
2 
3 
3 
4 
4 
5 
5 
6 
1 
2 
3 
4 
4 
5 
5 
6 
TT 
Y 
2 
3 
4 
5 
6 
6 
7 
< 
2 
3 
4 
5 
6 
6 
7 
8 
s 
V 
3 
4 
5 
6 
7 
8 
9 
9 TV 
5 
6 
7 
8 
9 
10 
10 
TT 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
1 1 
1 2 
I. Normal distribution for n even, =12. 
