i66 THE FACTORS OF THE MIND

where M is the initial matrix of measurements, suitably
standardized, L the orthogonal matrix of direction cosines,
V the diagonal matrix of factor-variances, and P the ortho-
gonal matrix of factor-measurements for persons. The
latter can be calculated by the equivalent equation—

P = V-*L'M.

If we decide to ignore the factors having the smallest
variance, on the ground that they have no statistical sig-
nificance, we are left with a set of measurements which gives
(for whatever number of factors is retained) the best possible
fit to the observed measurements as judged by the principle
of least squares. If the initial matrix M has been suitably
standardized, P and L can be deduced by correlating (or
rather covariating) either tests or persons : since, for the
covariances between tests we have Rt = MM' =LFL' ;
and for the covariances between persons, Rp = M'M =
P'VP (see Appendix II, Tables II and III, for worked
examples). Any of the standard methods for computing
latent roots and vectors can be used to calculate the factor-
loadings ; and, once obtained, these in turn lend themselves
to very simple algebraic or arithmetical manipulations for
deducing regression equations, factor-measurements, rotated
factors, simple structures, and the like.

We have now reviewed the chief rival theories hitherto
put forward ; and we may, I think, fairly conclude that all
of them—the ' multiple-factor theory/ the c three-factor
theory,' the c two-factor theory,' and the * single-factor
theory'—are merely special simplifications of the general
theorem of four factors. How many kinds of factors we shall
actually discover in any particular case must depend on
what traits and what persons have been selected for
examination. In general, and so far as psychologically
significant factors are concerned, the broader form of the
* multiple-factor theory '—a * theory of common factors,' it
might perhaps be termed—seems undoubtedly to provide
the best working hypothesis. Specific factors (in the narrower
sense of the word * specific '), together with the chance
factors of error, we may regard in the main as simply the