1 22 Proceedings of Royal Society of Edinburgh . [: 
II. 
1 2 3 4 5 6 
7 8 9 10 11 12 13 
0 0 0 0 0 0 
0 0 0 0 0 0<> T 
1 1 1 1 1 1 \ 
1 1 
2 2 2 2 *-g 
1 
2 2 3 3 3 
1 2 
3 3 4 4 \ 
2 
3 4 4 5 5 ‘V 
12 3 
4 5 6 \ 
3 
4 5 6 6 7 
2 4 
5 6 7 8 \ 
3 
5 6 7 8 9 
4 5 
7 8 9 10 1TT TT 
6 7 8 9 10 11 al T ^ 
1 2 3 4 5 6 
7 8 9 10 11 12 12 13 
II. Normal distribution for n odd, =13. 
We notice that when n is even the vertical middle column is 
JL l/y + 1 ) 
full : in fact, we always have the fraction f— or ' occurring 
\n \n 
in the class between — and , according as r is even or odd. 
n n 
And when n is odd the middle class contains, besides the limits, 
only the fractions numerically equal to J, for the number of these 
fractions is \(n - 1). 
§ 4. We also observe that in each case the last column contains 
one (i.e. \ + J) fraction in each row except the first and last, which 
contain 1J, and the second last column contains one in each row. 
We therefore find the following additional “ even ” distributions : — 
(1) Into n- 1 classes. 
Consider only the fractions */^> (n - 1 ). These are distributed 
\n in each class except in the extremes, which contain n-\. 
The fractions */n then fall one in each class except the extremes, 
each of which contains 2 : viz., between —— and — we have 
n - 1 n - 1 
— and — , between ^ and —^4 and so on. 
n n n - 1 n~\ n 
