1905-6.] Distribution of the Proper Fractions. 129 
restated in this connection. The evenest distribution for the 
assemblage */^j>w where the lower ratios */^>m are to be omitted 
is found by dividing into n + m+1 classes (§ 7). When the 
extreme fractions are counted \ there are then n-m+1 ) in 
each class. If n + m + 1 is even, pairs of classes may now be 
made to coalesce, and now reckoning the extreme fractions as 1 
there will be n — m + 1 in each class. This result is still true if 
the frequency curve is symmetrical from m + 1 to n. 
{Issued separately April 16, 1906.) 
PROC. ROY. SOC. EDIN. 
-YOL. XXYI, 
9 
