126 
Proceedings of Royal Society of Edinburgh. [sess. 
Then we can consider the n fractions */l to */n divided into x 
sets, */T to */m, */m + 1 to */2m, . . */(m- l)a?+ 1 to *jmx, 
and a partial set */xm -f 1 to */n . 
S 7* 7* -|— X 
Now if — is a fraction lying between — and , so that 
p m m 
pr p(r+l) 
— J> s j> > 
m ^ ^ m 
then between the same limits there will be t + 1 fractions */p + tm 
For let 
p + tm 
be such a fraction, then 
(p + tm)r 
m 
{p + tm)(r+ 1) 
m 
or 
pr 
m 
>*' 
-tr 
efc±D +< . 
s= tr + s + t\ (^ = 0, 1 , 2, . . t). 
But ii p^>m and no fraction *jp lies in this class, then 
s' = tr + ^ J + 1\ (f = 1, 2, . . .,t). 
Hence in the (t + l)th set we have each class exactly the same as 
the corresponding class in the first set, except for the addition of t 
fractions of each kind. 
It follows that if y = 0 all the classes will contain the same 
number of fractions, i.e. the fractions */%>n can be divided evenly 
into any submultiple of n classes. It follows also that they can 
be divided evenly into any submultiple of n 4- 1 classes, or any 
submultiple of n - 1 or n + 2 classes, though in the two latter 
cases the number of fractions in the extremes will be \ more than 
in the other classes. 
§ 9. The distribution into any number of classes m<n is easily 
effected. First write down the normal distribution for m. Then 
for the next set from m + 1 to 2m write down in each column p 
the numbers from 0 up to p consecutively, so that one number 
corresponds in each class to the number in the corresponding class 
in the first set, and there is one number in addition. For the 
third set there must be two additional numbers in each class, and 
so on. Table III. represents the distribution of the fractions 
*/^> 24 into 8 classes. 
