CLASSIFICATION OF METHODS 265

formulation a step further, and say that, given a suitable
set of observed variables, each of their r factors may be
defined by a unit hierarchy with a definite j actor -variance
attached. The final arithmetical solution of the problem
will be given by calculating the terms of the series

which may be called the factorial expansion of M : or, more
succinctly, by calculating the nr factor-measurements,
P = WM = 7-* L'M ([101], p. 84).*

In the pages that follow, I propose to show that all other
current solutions may be regarded as derivatives of this more
specific solution.

(v) Oblique Factors. — Throughout the theoretical part
of this discussion I shall as a rule use the term factors to
mean what are commonly called * orthogonal ? factors, as
distinct from ' oblique/ i.e. factors for which the factor-
measurements are uncorrelated or independent,2 and which
can therefore be represented by rectangular axes. The
relations between oblique and orthogonal factors are simple ;
and, for the 2- and 3 -dimensional case, will be familiar from
elementary geometry, (i) With oblique factors, the corre-
lations between factor-measurements and test-measure-
ments can be obtained by multiplying the matrix of factor-

1 For working instructions and procedure, see Appendix I . The reader who
is unfamiliar with the matrix notation used in the foregoing argument will
follow the points more easily if he turns to the elementary example worked out
in Appendix II, Tables I and IV.

2 In describing factors as statistically independent, uncorrelated, or
£ orthogonal,' most writers appear to be thinking solely of correktions between
the factor-measurements, and not of correktions between the factor-satura-
tions as well; but they do not state this explicitly. Thus, Thurstone's
preliminary analysis is based on the assumption that " the factors are un-
correlated " (p. 61) ; but the equations expressing this assumption show that
he is referring only to the rows of the factor-measurements in his * population
matrix * (P4). He postulates that " the n tests which constitute the battery
are so selected that the columns of the factorial matrix (F) are linearly
independent," and this is evidently meant to imply that they need not be
" statistically independent " (p. 57). With my * simple summation ' method
the factor-saturations usually have a low correlation, which may be regarded
as merely an effect of an imperfect mode of approximation. With the cen-
troid-summation method as used by Thurstone (highest correlation in leading
diagonal) the correlations may be appreciable.