2S8 THE FACTORS OF THE MIND

none but the statistically significant components will actu-
ally be calculated ; so that a comparatively small number of
the new components will account for far more of the
observed variance than would be accounted for by an equal
number of the original variables; (iii) nearly always
(though not in my view necessarily *) the function £ will be
taken to have the simplest possible form, i.e. to be a linear
function, so that we can write M = FP, where P is (by i)
an orthogonal matrix (or horizontal section of such a matrix)
and F a linear operator or matrix pre-multiplier. It might
seem reasonable to add, since reference-values are usually
needed for permanent reference : (iv) the set of components
selected must be stable for all samples of the categories from
which the n and N items are drawn, and for the same
samples on different occasions. That, however, is a result
which would require separate demonstration for each set,
and is, as a matter of fact, hardly ever explicitly established.2

and Kelley put r = n precisely. However, each of these writers recognizes
that all but a few of the numerous components deducible in theory are in
practice likely to be devoid of statistical significance.

1 Just as we should not assume a 'priori that both height and weight can be
expressed as linear functions of the same factor or set of factors, so, it seems
to me, we should not assume that mental performances must necessarily
be treated as linear functions of the related factors. It is interesting to note
that a good many of the familiar factor theorems do, as a matter of fact, hold
good with more complex functions; and, in any case, a linear function may be
regarded as supplying a first approximation to the more complex function
(cf. above, p. 241). Nor is it, I think, commonly realized that the product-
moment formula, on which our correlations and regressions are regularly
based, is merely the simplest case of a more general formula. Thus, in the

numerator we may write the generalized covariance - Sxty?, where f and q

11

do not necessarily = I, with corresponding expressions for the two gener-
alized variances in the denominator. The system based on these three
parameters should, in theory, enable us to specify the law of dependence
between x and y, no matter how complex it may be.

2 It might perhaps be contended that we only factorize tests or trait-
measurements that are known to be * reliable'; so that the reliability co-
efficients obtained (a) from different sections of the same test, and (b) from
applications of the same test at different times, would guarantee the stability
of the initial variables. Yet that hardly suffices to prove the stability of the
factorial pattern. It:would be better to include the separate applications or
sections as separate variables in the correlation table, and so show that the