DERIVATION OF CHIEF THEORIES 153

malized the vector of factor-measurements, p, so we may
normalize the vector of factor-correlations or * saturation
coefficients,' f1? thus expressing them as the product of a
* factor variance ' ^ and a single set of * direction cosines ' lx.
We may then sum up the single-factor theorem in matrix
notation as follows :

where H denotes a c hierarchy 5 or matrix of rank one,
E what I have called a c unit hierarchy ' [115], and l\ lt = i,
When obtained by the methods described below (Appendix
I, Tables I-III), the factor variance (v) and the direction
cosines (1) will be respectively identical with the latent
root and the latent vector of the correlation matrix.1

Three points may be noted in this theorem. First, it is not limited
to tables containing positive figures only. Neither the inter-
correlations, therefore, nor the saturation coefficients derived from
them, need be exclusively positive. This is of special importance
for several reasons. It is widely but wrongly supposed that the
most convincing ground for postulating a single general factor is
' the almost universal positive correlation among tests of mental
ability.' 2 Such an explanation is misleading. A saturation co-
efficient is a correlation coefficient ; and as such may in theory take
negative as well as positive values : only imaginary values are
excluded (the chief point which differentiates factor-analysis from
the corresponding procedure in quantum analysis). Hence,
provided the hierarchy satisfies the proportionality equation, half
the table (or rather two quarters) may consist of negative correla-
tions ; and the table itself will still be explicable as the product of
a single factor with real coefficients. The removal of this limitation
permits us to analyse bipolar correlation tables, such as result from
eliminating the first positive factor (as in tables of residuals) or from
testing a population that is homogeneous as regards the positive
factor, by means of repeated applications of the same single-factor
theorem. In consequence, when the observed table itself departs

1 I shall continue to use the symbol Vj for the latent roots, instead of the
symbol more familiar in matrix algebra, X> > because, in factor-analysis, the *th
* latent root ' represents the contributory variance of the rth factor — in the
present case the variance of the one and only factor.

2 Cf. Guilford, kc. «"*., p. 464.