1 64 THE FACTORS OF THE MIND

The Multiple-factor Theorem (Theorem of Superposed
Hierarchies). — Nowadays, however, every factorist, with
the doubtful exception of the strict two-factor theorists,
would acknowledge that we have to reckon with the pos-
sibility, and indeed with the probability, of a plurality of
common factors entering into most of our correlation tables.
Whether the supplementary factors cover all the tests, or
only a group of tests, or in rarer cases none at all or each
only one, must obviously depend upon the particular set of
tests we choose. But, in the broader sense, we are all mul-
tiple factorists to-day.

In this country the view is by no means a new one.
Already in 1917 I argued that we "have to recognize a
multiplicity of common factors," and proposed to meet the
more complex problem thus presented by invoking the
method of ' multiple correlation.' 1 On this principle, we
may assume that any n X n table of correlations or co-
variances can be expressed as the sum of not more than n
independent single-factor hierarchies. We may, in fact,
generalize the reduction given on p. 153 above, and write :

where #, as before, denotes a hierarchy (i.e. a matrix of rank
one), v the factor variances, and E the latent hierarchies.
If, as before, we use the * method of least squares,' the E's
will form a set of unit hierarchies, defined by the equations
1- I/ = ft. and L'L = /, where / denotes the unit matrix :

I
0
0
I
... o
... 0

o
o
... I __

(For a simple illustration, see Appendix II, Table IVA.)

1 L.C.C, Report, loc. cit. sup., pp. 53, 56. By the ordinary proof of
* partial correlation,' if I, 2, . . . n denote n independent common factors,
suitably standardized, we have r^ 112 . t § n = (r^ — fiI r^ — r^ rj2 — ...
— TfaTjn) -T- k ; and, if we assume that the final residual, r^.«... a1 *8
zero or approximately zero, we may write r# = sum of the products of the
paired saturation coefficients for the n common factors: this gives us a
formula identical with the so-called * cosine law ' (see above, pp. 88, 91).