154 THE FACTORS OF THE MIND

from the hierarchical form, it becomes possible l to treat it as a sum
of two or more hierarchies, the first or dominant hierarchy being
usually positive throughout, and the rest bipolar.

Secondly, the theorem, as thus interpreted, leads to a vast simpli-
fication of the data. We start with an n X N matrix, i.e. a table
of n test-measurements for N persons; on correlating, the product-
moment formula reduces them to a symmetric matrix of In (n — i)
covariances or intercorrelations (the n diagonal variances or self-
correlations being, in most cases, not independent data but arbitrary
figures depending on our units of measurement) ; then, if the table
is not itself a hierarchy, we can still regard it as consisting of a
dominant hierarchy (giving the closest fit to the observed figures
according to the principle of least squares) overlaid by one or more
relatively unimportant residual hierarchies; and, finally, this
dominant or best-fit hierarchy can be reduced by the single-factor
theorem to (i) a vector or column of n normalized figures only,
weighted by (ii) a single figure, representing the factor-variance.

Thirdly, the theorem implies that, if the figures representing the
test-variances or self-correlations (i.e. the figures in the leading
diagonal of the covariance or correlation table) also obey the
product theorem and so fit the general hierarchical pattern, there
will be no universal obligation to assume any specific factors at all.
This negative assumption was, indeed, the basis of my original
formula for calculating saturation coefficients by simple summation,2
On the other hand, even if the Inter- and self-correlations do not
obey the product-theorem as they stand, they can nevertheless be
expressed as the sum of sets of partial correlations that do obey it.

Can the Specific Factors be Dropped, ?—It will be observed
that my derivation of the hierarchy and my formulation of
the single-factor theorems do not include any mention of
specific factors. If, indeed, we regard a correlation matrix
as merely a special case of a covariance matrix, then no
specific factors seem required. I do not claim that specific
factors will never be necessary, but merely that they are not
always and automatically necessary*

1 A formal proof will be given later : cf, [102], p, 189, [115], p. 156.

2 See Appendix I, p. 474. The difficulty mentioned above (to discover
values for the variances) may appear even greater in the case of tables
which, for one reason or another, do not perfectly fit the hierarchy* As will
be seen from the account given in the appendix, the simplest device seemed
to te to insert figures that fitted the correlational pattern, and then if
necessary check their accuracy at the end (p. 448).