L. ISSERLIS 
401 
Now the mean values of a-y,. and ct-y,, are known to be 
o-,/ (1 - and 0-^" (1 - 
and the deviations from these are usually somewhat irregular. Rarely can we 
do anything better than assume them to vary with a slight linear variation from 
the mean. For example 
o-V,v = o-/ (1 - (1 + \x), 
where X is small. In such a case 
or to a fair degree of approximation*, we may put 
qxhj^ = 1 - r\,,, f 1 (/3.3 + ^82') r\y, 
and thus write 
^x2;/2 - = 1 + i (/^a + jSo' - 4) r%, 
= 1 + r^y + i (/S, - 3 + yS; - 3) r\,. 
The latter part of this expression vanishes if the frequency of the x and y variates 
be mesokurtic. It can of course be retained if desired but its product with 
xyH-z — xyRz will usually be of the second order. If we write 
t = i(A-3 + /S,/-.3), 
we find the approximate regression surface 
Z-z ^ r^^ - Ty^r^y {x - X) _^ Vy^j-r^zV^ (y - y) 
This equation (57) enables us to express approximately the multiple xi/H^ in 
terms of the simple yy^, ^Vz, xVy, nVx- 
I 7. To obtain this connection between the multiple xi/H^ and the simple y's 
we may proceed as follows ; 
Zxi, , ax bu cxu . . 
— - = d -\ 1- ~ H , origin at mean. 
Hence keeping y constant and summing for the a;'s 
^ = ^ + ^^ + ^ + ^ (.58). 
(Tz O-x (^xO'y 
R„, ., _ SSS [n^yz i'Zy - _ SSS (n^y, 0/) ^ 
0-2 ^'^'^ {>'X,/Z{^X,/-Z)'] SSS{nxyzZ\y} 
and xy-tiz - = , 
* I.e. we are neglecting terms of the second order as xV/Sj. 
