410 0)1 the Partial Correlation Ratio 
Let \ = fi =0 and either v or v' = 0. 
It follows from (79) that R = R, i.e. v = v'. ' ' 
:. both V and v' are zero. We have now four linear regression curves, .'. all 
six are linear. 
Let ix = v' =0 and ^' = 0. 
Sinee jju — v'^O, :. F = P' or A, = V, so that 
P =X\ Q = -<y.;-^X\ R = v\ 
P' = X-, 7, V, Ii' = -y./X\ 
(79) becomes (i^= + y^'A.^- ?%= 4r,,^73X,= ()-^.,^i/- - YaYa'A.-), 
(80) becomes 
(y.^^v- - y.^'X-y 7%, = 4 (Vy.X - r,.,iv) y-.y.^v- [{y.,v - 7/A.) + r^, (y.!v - 72^")]- 
Tli(^ first reduces to 
A.-70 (?■;,, + rff.rxy) r^y - 2 ry^ 
1 - 1% 
and the second to 
(y„y- - 7,'\-)= + 47,7./ (ry,\ - r^yv)- v" = 0. 
Hence either X = i' = 0 or there must be a very special relation between 
If instead of fjf = 0 we take v = 0 we get similar results, i.e. in general the 
vanishing fj,, v , or v, v involves that of \, v, X' or X, /u,', X'. 
This accounts for six more cases. 
There are eight left. Of these six are typified by 
fji = v = 0 = v or fjb = v' — 0 = fx, 
and lead to the same conclusions. 
The remaining two are 
X= fM = V = 0 or \' = /uf = v' = 0. 
The first supposition, \ = fi = v = 0 gives 
P = — yi'p'", Q = — 7/'"X'-, R = — y-i'/ji'", 
P' = \'"-, = R' = v', 
leading to 
(79) (i''' + 7:i'V)- r"y^ = ir^./x'-y-/ {r,ryv'"- - ysj^'/^'-}, 
(80) (X'= + 7i'^f'=)' '-V—^'-^^/^'Vl'V^'-TiTiVM 
(81) (a6'= + 7;'^X'=^y^ r\,, = 4jvX'=7/ {r^,/^ -. 727='^'=}, 
which give X'=/x' = i'' = 0 or a very special condition to be satisfied by the 
correlation coefficients r^y, Vy^, r^^x- 
