KmsTiNE Smith 
15 
Applying (25) and (20) we find for the mean values of the products under 
(24) the following values : 
{T(oc,y,)y- = nq [q {n + 1)7-/ + l+{q- 1) r] s's"- ^ 
(S (x]y\)y =nq(n-l) [qr^^ + 1 + (g - 1) r} s V 
t {x,y,) 2 (yi') = nq [nq +2 + 2{q- l)r} 7>sV 
2 (a;,y,) 2 {y,y^) = inq (r^ - 1) {2 + {nq + 2q - 2) ?•] />sV 
S («i2/i) S {y,y.,) = n (/? - 1) 3= {1 + (*7 - 1) r} j^sV 
2 («i2/i) 2 («i'^) = nq (n + 2) y^^s'-'s 
S (a-'i^/i) (a-'ia-'o) = n (n — 1) qr^s '^s 
■im. 
2 («i^) 2 (yi^) = nq (n + 2?-/) 
2 {x{') 2 (yi 2/,) = Inq (q-1) {2r/ + 7ir} s'-s" 
S {w.x^} S (y, y.,) = \n {n - 1 ) q-r/s'-s- 
We may now continue the calculation of W'xy Introducing the mean values 
in (23), we get 
n^q U%j = {n-l) {nqr/ +' 1 + ((/ - 1 ) r} sV. 
From (22) we find 
n^q (Uxyf = (n -If qr^'s"'s\ 
and when this equation is subtracted from the foregoing 
— — fl — 1_ 
<T\i,y = n\., - (n,,)^ = [qi-; +i + r{q-\)\ s^s^- 
(b) The Product Moment, Hn^^, o--, of Yl^y and a-. 
Multiplication of (4) and (21) gives 
n., . a"- = (nq - 1) (n -1)2 (yr) 2 (w,yO - 2 (n - 1) 2 (x.y,) 2 {y,y,) 
.(28). 
+ 2S(a-,y,)8{y,y.^ + y„ 
where 71 consists of terms S x 2, the mean values of which are zero. 
Taking the mean value and applying (27) we therefore find 
nY H-ry . a- = (h - 1) {nq {nq + 1) + rnq {q - l)j sV, 
For U^y . we get from (3) and (22) 
^^'tlUcy . = {n - 1) ?v [nq {nq - 1) - rnq {q - 1)} s's^ 
and accordingly from the two latter equations 
2 {n - 1) 
(29). 
