336 THE FACTORS OF THE MIND

how they are related to each other and to the more import-
ant alternatives that have been proposed. Let us begin
with the tetrad-difference criterion.

If we regard the correlation table as a matrix or determinant
arising from the solution of a set of linear equations,1 this criterion
can be reduced to a very simple form. As has often been pointed
out, it is—under a different title—the regular method for demon-
strating that a determinant is of unit rank.2 The great objection
to its use is the time and labour involved : with a dozen tests the
intercorrelations would yield 1,485 tetrad differences to calculate
(not, however, all independent) ; with Thurstone's latest table
[122] over a million. But when the determinant is symmetrical,
it is in theory unnecessary to compute all the tetrad differences : it
is sufficient 3 to show that the principal tetrad differences—i.e. the
diagonal minors of order 2—are all zero. The sum of these diagonal

minors, EE Tn ^ = (Er^ — EEr^f. If, therefore, the correlation

matrix is of unit rank, the sum of the squares of all the correlations
(inter- and self-) must be equal to the square of the sum of the
self-correlations, i.e. squares of the saturations with the single
factor : in other words, the standard deviation 4 of the correlations

should be equal to — X the total hypothetical test-variance. This

hypothetical total can be determined with sufficient accuracy by
the methods described in the appendix ; and the probable error of
the standard deviation can be estimated in the usual way. It is
curious to note that this form of the tetrad-difference criterion de-
pends on calculating precisely those tetrad differences which the
ordinary form of the criterion omits.

Since Era = Evt = Evf (where vt and Vf denote the n test-

1 Cf. Marks of Examiners, p. 247.

2 The * tetrad ' is a two-rowed minor determinant renamed ; and the
' tetrad difference ' its expansion. Hence the criterion may be regarded as a
special application of the familiar property of determinants, namely, that
* if the corresponding elements of two columns (or rows) are proportional,
the determinant vanishes/ For the criterion as a theorem in matrix algebra,
see Cullis [26], 1918, II, pp. 93, 139; and cf. Bocher [15], 1907, p. 34 f.

3 Strictly we ought also to show that the diagonal minors of order 3 are also
zero. But this is needed only to rule out certain patterns involving negative
signs which are never likely to arise in psychological work.

4 The c unadjusted' standard deviation, if we are dealing, not with a
bipolar matrix of residuals, but with an observed table of positive correlations.