H. E. SOPER 
387 
Mean So is to be evaluated by the formulae 
mean 8p„Sjj„ = {'Pu+v - puPv)!^ \ 
mean hp^hp^ = {p\t+v- Pu Pv')IN \ (16), 
mean Sj3„S|V ={pu+v-PuPv)IN j 
of which the first two are well known* and the third may be proved thus: 
Nhpu = S = >S' (an,'?/..") + S (Bn/'y-), 
NSp^'^SiWys"), 
.-. N-'BpuSp; = S{{Bn:yy,^^+'>} + S{8n:8n/y,^hj,«}+S[8u;^ 
where in the third sum s may or may not equal s. 
But mean (8n;)= = ?z/ ( 1 - n^'/N), 
mean (Bng'Sn/) — Hgn/jN, 
mean {Bn^'Bug'") = — nJus'/N, 
the last whether s = s' or not. Thus we findf 
mean 8p„ = (p'lc+v - p^ipJ - pu pv ) I N = {p'u+v - Pupv')/ ^■ 
Evaluating mean S.^ by these formulae we find, 
4^P,'-.,K)-i(K-KP,) + J/(ft-,„.. 
Z 
- (Pi -Po'Pi) + i^'' (P-^ - pi') + 'I ilh-pi') + (1 + 2«^) £ {p^-jC)^ . . .(17), 
in which the undashed moments, being those of a normal curve with unit standard 
deviation, about its mean, have the values 
Pi = 0, Pi=\, Pi = 0, ^4 = 3, 
and the dashed moments beyond the first two, 
P<! =./', Pi = zr, 
have values depending upon the nature of the frequency distribution of y and x. 
Assuming x, y normally distributed J 
* See Biometrika, Vol. ii. (1903), p. 275: "On the Probable Errors of Frequency Constants." 
K. Pearson. The second follows in exactly the same manner as the first, since the constancy of the 
total frequency dealt with is only involved, in deducing the relations (i) and (ii) of p. 274, of that 
paper. 
t p," = S(n,"y,")jN. 
J Since the moments appear in the term containing 1/2^ any errors in their calculation due to 
incorrect assumption of normality will not affect the present approximate formulae provided such errors 
are of the order IjN. 
