Karl Pearson 
127 
of r by fourfold — would be far safer to use than r c q , or any arbitrary coefficient 
of association, which wholly neglects the question of the type of the frequency 
distribution. In the rare cases in which the frequency is collected into points — 
absolute homogeneity of category contents — then fc s C y * s the right coefficient 
to use and not Yule's coefficient of association. 
(9) I propose in this section to consider the relations between 
8 
n st - 
N 
x st ■ JJst 
s 
n st - - 
) and S r~ . x s . y. 
when the subranges for each character are small and equal. Let those for x be h, 
and for y be k. Then consider the surface 
z = a + bx + cy + dx- + ey- , 
and choose a, b, c, d, e so that it gives by its volumes correctly the five class 
frequencies 
h—l,t> "s,t< "s+i, t> "s, t—n 
S, t + 1- 
Then if we take the origin so that it lies at the mid-point of the n Si t group, 
we find 
2 = Ik { ns ' 1 ~ 12 ( n *~ 1 ' 1 + n *' ~ 2ns ' d ~ ii ( n ° +1 ' ' ~ n *- 1 ' 1 + ,+1 ~ " s - 
, 1 * t - n s -i, t \y n s , t+i - t-i 
1h hk 2 k hk 
+ 
« 2 f n s - h t-n s ,t M th+ h t-i\ t \ M if f 'h, t-i - » g , t , - n s , t 
Further 
hk + 2hk 
r+Uc r+hh 
J -hk 
+ — I 
hk 
2hk 
^xzdx = ~bh"kJ~(^ 
Irk 
Or 
Similarly 
1 7 n». t+i ~ n s, t-i 
y s t = y t + ^ 
.(xxviii). 
..(xxix). 
Now take the product of these, multiply by n st and sum, we have 
8 $ x st y st ) = 8 x s y t ) +^Mf{jf (n. +J , , - n s _ h ,)} 
+ 9l kS fe^M-H ~ n s, t-l) 
+ hkS 
144 
n^t-n — n Sl t + v g ,t-n s 
2N 
2n- 
But if h and k be small the last summation is, precisely as that dealt with on 
p. 122, of the fourth order, and we have . 
