XVIL.—NORMAL CORRELATION. 219 
tribution of each of the deviations singly be normal, we must have 
for the frequency-distribution of pairs of deviations of x; and z,., 
Vig =Y10 gy 93) A 
IER 
xf | 3, a; x; o Ty 
St = Frag my ni 
or 031 oi(l—-1}) oy(1 —1},) oyoa(1 — 77) 
2 2 
Ba 2. 
PR 
Rk Ojot-05y 912021 
Evidently we would also have arrived at precisely the same 
expression if we had taken the distribution of frequency for z, 
and z, ,, and reduced the exponent 
Oz: Oi» 
We have, therefore, the general expression for the normal 
correlation surface for two variables 
2 2 
x x 2s 
SC NE. (6) 
’ a go, 2 : 
Yi2= Yat 1.2 21 1.2 21 
Further, since #, and ,.,, z, and 7.9, are independent, we must 
have 
ie YW h . 
Y= 27.0100; 27.0000, 2mo, op(1 — ri) - (7) 
4. Tf we assign to x, some fixed value, say h, we have the 
distribution of the array of x,’s of type A, 
( = Le op i ) 
hy ols oh; of 2721 
Y12=Yr-e : 
a1 z 
‘2 (= lt) 
= Vine 3 207 
This is a normal distribution of standard-deviation 01.0 With a 
—r o Tp 
mean deviating by r,, hy from the mean of the whole distribu- 
2 
tion of z’s. As A, represents any value whatever of z,, we see 
(1) that the standard-deviations of all arrays of x, are the same, 
aki : 
(v 
Buu ’ 
i