406 
On the Partial Correlation Ratio 
In the important case /3i = l3i'=0 and = ^.7/,, = these approximations 
fail, and so does the process by which (70) was obtained as 6 and </> vanish and 
(61) is indeterminate. 
We must then fall back on equation (59) 
rcy Hi - y ?// = + C-) ( 1 - „ 
= (7-/ + C-) (1 - ?■%;) since ^ = 0 and = )\.,, 
- ' >/) + ■ s — (1 + «a). 
^.r^i/^ ~ l"xy 
or (,,„ H;- - Ef) [l- ~ 4^ \ = 1^;' - + 73- ( 1 - 1%), 
neglecting 3rd order terms. 
The right-hand side reduces to yVz" ~ ''V> ^'^'^^ 
.r,jHi - ,„R/ = nifH^:^ (,^^^. _ (71) 
". ' 2r^ . 
= z ^' (,,?// — r-2,,) approximately ; 
1 "I" ^'"xy 
of course in these circumstances (60) would lead to the value 
^!(xvJ'->■^^ (72), 
^ -r ^ xy 
showing that if = /3,' = 0 and if (xVy^ " ''"^w) — (yVx^ - '"'.rv) is of higher order than 
the first then (.^77/— r'rx) — (yVz" — >''zii) is also of higher order than the first. 
We shall now seek relations between the six correlation ratios of three 
" hyperbolic " variates. 
From (.5!)) and (60) we get, on eliminating xyHf, 
yVz^ - xVz" = h-- + c- (,,7?/ - ^rjy'y + a\jr)/ - b\7}y' 
^y-/^-jI + y/yVx'-y-i'~xVy+^(r.i0yV.T-'y3 4>.rVy'') 
= (73'' - 7:') (1 - + 7/ (yVx^ - r-xy) - 73'^ (xVy'' " ^'xy) 
+ {2c73^ {y7]^- - /%) - 2073'^ (:,77,,- - r\y) 
+ {0- - <})- + yrj^^ - xVy + &\jVx^ - (f^'xVy")}- 
The terms in the second line are second order terms. Neglecting these and 
noting that 
(7/' - - >''xy) = ''V - ''"'xz and (73^ - 7.,» (1 - r%) = Vy^q^y. - r^^q^iy, 
we obtain the following equation for c : 
{yVz - r-zy) - (xVz^ - r^x) - 73' (yVx- - r%) + y-/- {xVy^ - 
= 2c (r^zqx^y - ry.q^y,) (73). 
