332 THE FACTORS OF THE MIND

actually states : " fortunately the diagonal entry may be given any
value between zero and unity without affecting the results
markedly " (p. loS).1

We may, therefore, fairly conclude that, in principle,
the saturation coefficients given by the centroid equation are
virtually the correlations of those tests with the unweighted,
average of their scores, and that, if there is any slight differ-
ential weighting, it will depend solely on the somewhat
arbitrary figures chosen for the communalities, and in any
case will not " affect the results markedly," except when the
tables are small.

The factor-measurements obtained by the method of
least squares, on the other hand, are weighted averages.
But there will be no need to apply the principle of least
squares afresh or to calculate the ratios of the resulting
determinants : for we now have F'R"1 = F~1F''. Thus
we have now merely to divide each column of saturation
coefficients by the corresponding factor-variance, and we at
once obtain the appropriate weights.

pure hierarchy with unity in the diagonal cells is also of the same quasi-
triangular type, with nearly all the peculiar features illustrated above in
Table IIIB. In both examples it could be shown that the factorial matrix
obtained with the one method could be rotated to give that obtained with
the other by a quasi-triangular transformation matrix of the kind described
on p. 305. As will be seen by considering its special features, when the correla-
tion table is very largey this transformation matrix tends to become a unit matrix
with a column of units prefixed.

1 The student who does not follow the above proof by matrix algebra can
satisfy himself of the equivalence either by studying the tables below
(pp. 391 f.) or by following the inverse procedure. Let him first calculate the
factor-measurements for g by summing the standardized measurements for all
the tests, and correlate these sums with the measurements for any one test :
the correlation of each test with g will then prove to be identical with its
saturation coefficient for g (see [101], p. 75). If he likes, he can now generalize
this algebraically by using the formula for the correlation of sums ([47],
p. 197, eq. 147, putting a = l).

It may be noted that we have here a method of reconciling Thomson's
sampling theory with Spearman's two-factor theory : for we have only to
take the sum of the test-measurements to be expected in accordance with the
sampling theory, and correlate these sums with the most probable measure-
ment for any one test: we then find that these correlations have the same
properties as Spearman's saturation coefficients as deduced from a pure
hierarchy. So that once again the sum of the test-measurements has the
properties of g and vice versa.