120 
and 
Influence of "Broad Categories" on Correlation 
_ 2 _ „ n,._! - Wg+ i h?_ {n s -i - n s+1 ) 2 
, h , , h 2 h 2 It 2 (n^ - n s+1 )- 
= n s x s - — Y2 v r s-i"s-i _ x s+i n s+i) — |2 n »- 1 — 12 >ls+l 576 n 
Summing for all classes and dividing by Na x ? 
J? a 
g >h ®s 
S (n & x 2 ) 1 ^__^J^_ M & q ( n s -i ~ n s + iY 
No 
12 
12 
576ov 
But by Sheppard's Theorem, with contact at the tails of the order we are 
supposing : 
S (n s * s 2 ) 1 , , 
*■ =-ir~ "i2 /r - 
Further, if we suppose y s -\, y&> ys+i> Vs+2 to be the bounding ordinates of the 
classes n^, n g , n s+l of the frequency curve, 
= - y s h = - (-^ + 5^ + 2^) = 1 ^ _ ^ + ^ 
Similarly « s+1 = n s+J - y s+2 h = 1 (n^ - 5n s + 4/i s+1 ). 
Here ot s _i and a s+1 are the excesses of and n~ above rectangles ; and we have 
a s _j - a s+J = \ (w s _! - n s+l ). 
Thus finally 
.(xiv). 
W«r«V 12<r» + 14k« JVn s 
Consider the last term ; it may be written 
ji (a, - a s _, + a s+1 - a s ) ^ J (a g - a 4 _j + a g+1 - a s ) 
N 
-y x 
h- 
3 Gcr,. 3 
Ah - 1 
WMIIIIItMMmimmm*. 
2/8-1 
-><e 
Now the numerators are mean differences of what will be, if h be at all small, 
small areas, and these mean differences are divided in the one case by n s and the 
other by N, much larger quantities. Hence the last term is as a rule very much 
smaller than the second, and we have approximately 
