60 
On, the Partial Correlation-Ratio 
It is easy to remove this restriction. As on p. 402, we write 
= «M1 - )+ 2«c (1 5^ I) + c^^, |) (59)' 
/2 ^2 
If the a; arrays are honioscedastic = 1 ~ x^^v ^i^d we get (59) 
Similarly equation (60), corrected for heteroscedasticity, becomes 
B^ - rf. ^ 6M1 - rf.) + 2fcS„ (£ I) + c^S. (^^_ ^ I) (60)-. 
Applying these corrections to the case of x,jH^z, we find 
^,H% - -724535 = -0221969 + -0003345 + -0006875 (59)', 
or ,,H\ = -7478 
and ^yH% - -713674 = -0424845 - -0053862 + -0010073 (60)', 
or ,yH% = -7518, 
showing the separate contributions of the various terms. 
The agreement of (60)' with the empirical value -7519 is remarkably close. 
For the regression of weight on height and age, equations (36) to (39) give the 
values 
a = -24570, b = -70004, c = -12160, 
if the regression hyperboloid be written in the form : 
^'^ ^ = d + a + b h c ^ '-^ ' . 
(^V f^z CTx OTx ^z 
The values given by the equations corresponding to (59) and (60) are 
,^H\j = ,rq\ + (a2 + c2) (1 - ^7]\) = -838012 + -021519 = -8595, 
, = ,7j\ + (62 + c2) (1 - ^r^\) = -693989 + -15580 = -8498, 
while the equations corresponding to (59)' and (60)' give 
,^H\ = -838012 + -0172780 + -00011322 + -0034254 = -8588, 
and ,^H\ = -6939891 + -15128 + -0048971 + -0091002 = -8593, 
the empirical value of ^xH^y being -8634. 
These approximations are not quite so good ; they are bound up rather closely 
with the assumed hyperboloidal form of the regression surface, while Mr Soper 
has shown that in the case of weight on age and height, a good fit is obtained by 
a surface of the form: 
T/zx = «0 + + ^2^^ + (^0 + f^l^ + ^2^^) «*• 
8. Mr M. Greenwood, Jr, has kindly drawn my attention to a joint paper, 
entitled "A Study of Index Correlations," pubhshed by Mr J. W. Brown, Miss 
Frances Wood and himself in the Journal of the Royal Statistical Societyf. 
* Elderton's paper, loc. cit. p. 294. 
t Vol. Lxxvn, Part III, February, 1914, pp. 317-346. 
