128 Proceedings of Royal Society of Edinburgh. [sess. 
for p= 1, 2, . . \(m- 1) or \(m- 2), according as m is odd 
or even. The last term on the right will of course vanish if 
(x+\ )m-p>n, i.e. >mx + y, or if p<m-y; and the last term 
on the left will disappear if p>y. Neither will occur if y = 0; 
both will always occur if y = m - 1 . 
These equations are satisfied if y p = y m - P > . • . , p p + m 
= g 2 m - P , . . • If y = 0 or m - 1 we have a complete set of pairs, 
and none of the fs vanish, but in other cases certain of the /z’s 
will vanish. Thus for y — 1 or m — 2, p n = 0. 
Hence the distribution into m classes, m a submultiple of n or 
n+\, will still be even if the frequency curve for the denominators 
is periodically symmetrical with period m. 
§11. Let us consider now the effect of omitting certain of the 
lower denominators. We may for this purpose consider the 
assemblages */ */ ( xm - 1 ), */^> (xm + 1 ), */ Ij> (xm - 2), 
each distributed into m classes. 
The omission of a complete set or any number of complete sets 
will not affect the distribution. 
The omission of one less than a complete set or any number of 
complete sets will not affect the distribution. 
The omission of one more or two less than a complete set or 
any number of complete sets removes J more from the extremes 
than from the other classes. 
The results are indicated by the following table : 
These results on submultiple division require considerable 
modification as regards the extreme classes when the extreme 
fractions are reckoned 1 and not J. The number in the extremes 
is then not even approximately equal to or a multiple of the 
number in the other classes. 
§ 12. One result which we have already obtained may be 
*/ xm and */ ^> (xm + 1 ) and 
*/^>(xm- 1) 2) 
Extremes \ less. Same in each. 
Same in each. Extremes J more. 
