GENERAL- AND GROUP-FACTOR METHODS      307
Matrices approximating to.the curious triangular form illustrated
by Tables Ills and IIIc are by no means uncommon in factor-
analysis. I have already ([93], p. 307) drawn attention to their
frequent appearance, and have indicated how they are related, as
regards algebraic origin, to the triangular matrices that appear in
the canonical reductions described in nearly every textbook, and as
regards logical significance to the triangular tabulations that result
from repeated classification by dichotomy. The principle, I venture
to think, might be more directly exploited for practical calculation
as well as theoretical interpretation.^1
(c) Relations between the Factor-measurements.—Nor is it
true that the factors originally reached by a general-factor
method of analysis are necessarily devoid of meaning, or
that, ^c no matter how the correlational matrix is factored,
the axes must always be rotated before any psychological
interpretation can be made' ([16], p. 74). If we grant a
psychological meaning to the simplified factor-pattern
obtained after transformation, then (as I think will now be
^1 Let me give an instance in which much the same principle may be applied
to a situation not infrequently arising in actual practice. Suppose, for
example, one of our tests is an almost perfect test of g = s_1 (say). By the
repeated application of an obvious modification of the formula for partial
correlation :
w- 4?^1,.. ....
*= 1,2, . . ., a— i)
we can eliminate first g and then (n — i) other factors in turn thus obtaining
saturation coefficients for n orthogonal factors: these saturation coefficients
will remain unchanged, even if further tests are added to the battery. Where,
as so often happens, our tests show a cumulative complexity, this form of
analysis may well be employed, though (for reasons given in the paragraph
cited) it seems unsuited for correlations between persons. The procedure is
equivalent to a Lagrange transformation—the * rational reduction of a
quadratic form' to a sum of squares by a succession of non-singular linear
' transformations^? (for proof and arithmetical illustration, see [15], p. 132 ;
cf. [73], pp. 85-6). The cumulative transformation matrix is then triangular;
and is admirably adapted to express partial or cyclic overlap. If, on the
other hand, the specific factors do not overlap at all, so that there are no
group-factors, the resulting zeros convert the triangular matrix into the
hollow factorial matrix (a simplified * prop ladder') which, on removal of the
perfect test, represents the Spearman two-factor theorem. Thus we may
regard the solid triangular matrices as produced by a filling-in of the Spearman
factorial matrix to allow for overlapping.