L. ISSERLIS 
59 
1/2 _ P2 = _ '__L^" ( „2 _ ,.2 ) 
xv'-^ z xy-'-^ z o,.2 \y'l z ' zij]i 
In this example also as J^^ is much nearer to zero than ^/S^ we get better values 
from (72) and (72)' than from (71) and (71)'. 
We may conclude from these numerical cases that, if 
(i) ^(8i and ^jSj are fairly small, and 
(ii) the regressions of ?/ on a? and that of on y do not depart widely from 
Hnearity, we get very fair values for ^yll'^^ from an equation of type 
1 
and a very good value from one of type 
772 _ 7?2 = ga'i/- ~ ^"a 
xv'-^ z XV"' z 1 
Ix^v"' 
and that it is better to use the equation which corresponds to the smaller than 
to use the mean of the values given by the two equations of each type, i.e. if ^^-^ 
is smaller than we use the formula in terms of ^7y% — r^^y in preference to the 
one involving ^yf'^ — r^^, and in preference to the mean of the two. 
If the conditions (i) and (ii) are not satisfied, we use equation (70), which requires 
more labour than (71) or (72), but is much easier than a direct calculation. 
7 *. It is a matter of interest to see whether the regression surface of age on height 
is of the type considered in Part I of this paper. The regression surface being 
Zxv-'^ (jjx- _^ h{y -y) ^ c {x - x) {y - y) 
e {x - xf f{y~ yf . 
+ + 
The solution of equations (29)-(34) is : 
d = -060374, c = - -0940467, 
a = -360742, e = -072020, 
b = -547960, / = - -040709. 
Here c is decidedly smaller than a and b, but e and / are of the same order 
as c. This seems to suggest that our formulae have a wider range than the type 
of surface by which they were suggested. If we solve equations (36) to (39) 
(Part I) in which the coefficients e and f are assumed to be zero, we obtain 
d = -060172, b = -512257, 
a = -376024, c = - -066306. 
On substituting these values in (59) and (60), we find 
,,H\ = -7470, 
or ^yH\ = -7534. 
Now equations (59) and (60) were based on the assumption that the arrays of 
y's for constant x and the arrays of x's for constant y are homoscedastic. 
* This section differs from the preceding — it does not deal with genuine approximation formulae — • 
for in the determination of the constants a, b, c, d we use qx,j-i, i-C. the detailed table. It illustrates 
some other points in Part T. 
