CLASSIFICATION OF METHODS	261
mutually independent, statistically significantlinear components,
derived from an observed set of measurements by a homogeneous
linear transformation ; and each particular factor may be
defined by stating its correlations with the several tests as
specified by the corresponding column in the matrix of the
inverse transformation, namely, F. In geometrical language
the resolution of the test-measurements into factorial com-
ponents may thus be very simply described as consisting in
a change to orthogonal reference axes.
We can put this in a more concrete way. Since M = FP, the
;th element in the rth column of M (e.g. the ;"th measurement for
the zth person) can be split into a sum of weighted measurements,
r
mji = Sfjkpki = \mji + 2^mH + • • • + titty say.^1      Here   the   kth
figure, k^mji, is attributable to the kih factor only. On covariating
the kih figure for all the persons, we obtain f_Ap& X pa%' = fjfef/, a
symmetric matrix of rank one. Thus, we can summarize our
algebraic interpretation in words, and say : a factor is the class of any
set of variables, including parts of observed variables, whose covariances
form a perfect "hierarchy. For example, if we imagine a set of
n fictitious tests and if we suppose that the \n(n — i) relations
between them can all be expressed in terms of n constants only, one
constant for each test, then all those tests would measure one and
the same factor, i.e. they would all belong to one and the same
irreducible class.
Even so, however, the operations specified by the definition are
not determinate. No actual tests are likely to satisfy precisely
these exceedingly simple relations; and in general, an empirical
covariance matrix, £, can only be fitted exactly by a factorial
matrix F containing n* figures. Yet, in virtue of its symmetry,
-R contains no more than \ n (n -f- i) different figures, and so yields
at most \n(n -f- i) equations for the purpose. If we confine our-
selves to significant components only, we shall require fewer figures
than #^2 : but this general instruction is too vague to yield a unique
procedure. Unless, therefore, an investigator tells us what particu-
lar method he is adopting to obtain determinate values for F_9 we
cannot say precisely what he understands by a factor.
(iii) Doubly Orthogonal Factors.—So far I have followed
the ordinary approach to factorial work. " The object of
^3 Of. Appendix II, Table IVb.