120 Proceedings of Royal Society of Edinburgh. [sess. 
Suppose first that s + s >r + 1. 
Then 
n r+1 r+1 n 
which is impossible. Therefore s + s' r + 1, and similarly 
s + s' <^r. Hence s + s' must he equal either to r or to r + 1. 
If s + s' = r, 
n r 
nr r ’ p n n-p 
Similarly, if s + s' = r + 1, -IS y — - = — - — . 
p n n —p 
It follows from this that each class must contain either */P or 
*/n — p, and if any class contains both , each of them occurs 
counted ^ ; for, leaving out the denominator n, there are n — 1 
denominators, and only n - 1) fractions in the class. 
§ 3. The classes in which any fraction * jp occurs in the normal 
distribution can be easily found. 
Divide n, 2 n, 3 n, ... by p ; let q 2 , q 3 , . . . be the 
quotients, and f v f 2 , / 3 , . . . the remainders. 
mi sn f s 
Then — = q s + — • 
p p 
j sn .. , , 
>?»+!. 
n p n 
Therefore the fraction — lies in the (q s + l)th class. If f s = 0, 
— = q„ therefore = so that the fraction — lies A- in the 
p * pn p 2 
g s th and \ in the (g s + l)th class. 
Also, if n, 2 n, ... he divided by n-p , and Q p Q 2 , . . . 
are the quotients and Y lt F 2 , . . . the remainders, then the 
Hence 
or 
