L. ISSERLIS 
403 
i't'yz ~ ^'wz'>\y) {'''xylx-y ~ <lxy-) yVz _ i^xz__ '>'yt'>~xy) j'l'xy^xy- <lx^)xVz 
] (1 - r\,jy (1 - y-n,^) ' (1 -y-^.yfO-yvJ) . 
(I'xz ~ fyz^^xy) (fyz ~ ''xz'^'xy) i'^yzQx'^y ''^xzQxy- ) 
' i^-r'xyf 
+ 
xy^''z 
D 2 ~ '''x z>'xy) ( l'xy(lx-y ^xyd __ i'^'xz I'yz^'xy) (Xxy^xy- (6'^) 
: In general the square of any correlation ratio (ordinary or generalised) differs 
little from the square of the corresponding correlation coefficient ; also we have 
seen in (55) that an approximate value for qx-iu is 
rxy V^i - '^xV,'' - r\y V/3o - /3i - 1 . 
Now /8i is itself in genei-al small, so that without making any assumption as to 
the relative order of magnitude of /3i and xVy' - '""'xy we may safely treat qr^iy and 
qxyi as small quantities. 
Thus it appears that (62) is an equation in which all the terms involved are 
small, and a certain amount of care is required in deducing from it an approxi- 
mation to the value ot\yHf — ^.yR^-. 
We have 
t . = L_ _ ^ }L!k ''1m = ^_ + say 
1 yVx' 1 '^'^xy (1 yVx^^ (1 ^^xy) 1 ^'x;/ 
1 1 Tl 2 _ 9-2 
^ ^ X 'ly ' xy 
— + 
1 - x'r]y 1 - r-^y ( 1 - ,.?7,;-) ( 1 - r\,J) 1 - r'-xy 
.(63), 
1 - 1 
1 ^ '•\-// 
1 - ''"x// 
.(64), 
where X _ '>"zy( y Vx^ 1"xy ) j yVz' ^" zy .^^j ^ _ (yVz'^ f'^zy) (yVx' 1'^xy) 
(1 -//W)(1-^'V;/) 1 - ' ' 0^ - yVx-) {'^ - r\y) 
■m, 
while = +X/ + V 
where X,', \.! may be obtained from A,, Xo by an interchange of x and y in the 
suffixes. 
^ ^ + Af2 (65), 
Finally 
where 
(Ix-y- ^"xy ' 
Q~xy- + <)\-y ~ ^(jxy-' jx'^yfxy ^Jx'^y^ ~ 
1 - r\y 
Q'xy- ~t" ^f'x-y ^^xy-^x'^y'^'xy 
xy 
(1 - ^%/) (9.T=,/2 - r\,,) [q^iyz - r\.y - 
q%li + <fx^y — "^qxy^^iyrxf. 
The suffixes of ^/. X,, Xo, X/, X/ and /c, denote the order of smallness of 
these terms. 
