XIL—PARTIAL CORRELATION. 205
the coefficients 6,4 . . . . , etc, and so on: they are sometimes
termed the normal equations. If the student will follow the pro-
cess by which (5) was obtained, he will see that when the con-
dition is expressed that 4,,,, _ _, shall possess the “least-square ”
value, z, enters into the product-sum with #4; ... .,; when the
same condition is expressed for d,5,, .. ., ; enters into the
product-sum, and so on. Taking each regression in turn, in fact,
every x the suffix of which is included in the secondary suffixes
of #05... , enters into the product-sum. The normal equations
of the form (5) are therefore equivalent to the theorem—
The product-sum of any deviation of order zero with any deviation
of higher order is zero, provided the subscript of the former occur
among the secondary subscripts of the latter.
8. But it follows from this that
(2134... 234...n) =ZZ1s4...n(@p—boss...n.Tg— 0. —Donss... n=1. Tn)
=3(21.34 ...n. Tg)
Similarly,
(21.34... n. T2354... n) = Z(x, «X34... 0)
Similarly again,
Z(@134...n- T2354... (0-1) = (L134... mn. Ty),
and so on. Therefore, quite generally,
2(Tray nea n=3(x , ... n=1)+®234 ....m)
= Tyg... nm)
= BE «0 ETSY 3 irs nerd)
=2%15... n-&)
Comparing all the equal product-sums that may be obtained
in this way, we see that the product-sum of any two deviations is
unaltered by omitting any or all of the secondary subscripts of either
which are common to the two, and, conversely, the product-sum of any
deviation of order | with a deviation of order p+q, the p subscripts
being the same in each case, is unaltered by adding to the secondary
subscripts of the former any or all of the q additional subscripts of
the latter.
It follows therefore from (5) that any product-sum is zero if ali
the subscripts of the one deviation occur among the secondary sub-
scripts of the other. As the simplest case, we may note that a, is
uncorrelated with z,;, and z, uncorrelated with z,,.
The theorems of this and of the preceding paragraph are of
fundamental importance, and should be carefully remembered.
Rf
