260 THE FACTORS OF THE MIND

test of stability : only then it tests, not so much the psychological
significance of some particular factorial pattern in the light of our
preconceived notions, but the stability of our preconceived notions
about the factors in the light of yet another empirical table.

On the other hand, as I have argued elsewhere, a rigid adherence
to the postulate of invariance would force the factorist to surrender
(a) the idea of specific factors, peculiar to a single test or a single
person, (I) the idea that the factors obtained by correlating persons
are essentially different from those that are obtained by correlating
tests, and ultimately, I believe, even (c) the idea of standardizing
scores on the basis of the given sample. This postulate, therefore,
rules out so many ideas still widely favoured by factorists that it
seems hardly proper to include it at the very outset in the formal
definition of factors in general.

To begin with, therefore, it will be best to introduce only
the first three of the four restrictions mentioned above.
And It would perhaps be convenient to keep the term
' factorsJ for a particular and determinate set of compon-
ents, expressly selected so as to conform to these conditions,
and to use the term ' components? for * factors' in any
broader sense.

(ii) Orthogonal Linear Factors.—If we accept these
three conditions, we can take a further step. For simplicity,
let us keep chiefly to the type of problem for which factor-
analysis has been most commonly employed, namely, the
testing of n traits for N persons. As a rule, the sample
population tested will not only be comparatively numerous
(N > n), but will also vary from one inquiry to another;
the tests will not only be fewer in number, but, if duly
standardized as regards material and procedure, will
presumably remain constant. Hence it seems natural to
define our factors in terms of the tests alone.

Now by matrix multiplication (a common device in'
matrix work for reducing a given set of figures to a simplex;
form) the N individuals can be at once eliminated. We
have R = MM' = FPP'F' = FF'9 where (assuming M to
have been suitably standardized) R is the n x n matrix of
correlations, and F is a matrix of factor-loadings of order
n x r. Accordingly, incorporating the restrictions stated
above, we may now define our factors as a hypothetical set of