KiRSTiNE Smith 
5 
We find then 
=\nq{q-l) 
My^.¥i) =lnq{q-l) (q - 2) /3,3„ 
2 2/32/4) = 4^nq {q-\){q- 2) {q - 3) Y 
I '/i (n- 
S (2/l^!/22/3) ■-= i {n -l)qHq-l)/3,,, 
ds 
S (2/1^22/32/4) = in (h - 1) ^2^? - 1)'-' A 1 1 1 
Till now no suppositions have been made as to the law of distribution of the 
y's, but m the following calculation we shall suppose that tite distribution is normal 
and the correlation between individuals of the same class normal. 
For the general case of normal correlation between n variables the product 
moments have been determined by Sverker Bergstrom *. Taking the standard 
deviations as units of the variable and denoting the correlation coefficients by 
?-i2, r^..., where for instance r'as means the correlation coefficient between the 2nd 
and 3rd variable of a product moment /S'ntnpq, he finds the following formulae for 
the product moments of the 4th order : 
/s; =3 \ 
/S'k =2rS, + l i (9). 
/S'aii =2ri2ri3 + n., 
/3 mi — '''l2''34 + ''l3''24 T ''14 ''23 ' 
Substituting our special values for the correlation coefficient we find 
13, =3s^ ^ 
Ai =3rs^ 
= (2r"'+ l)s* 
= )•(! + 2?')s* 
= 2r-s' 
and further 
A2 
/^iiii 
(10). 
/32 2 =s* 
d 
^3 1 1 = rs' 
d s 
iSi 1 1 V 
s 1/ s 
We are now by means of (8) and (10) in a position to evaluate the mean values 
of the products put down under (6) and (7). 
* Vide S. Bergstrom : Biometrika, Vol. xii. 1918, p. 177. 
