366 THE FACTORS OF THE MIND

tions (or covariances) between persons as well as between
traits.

(a) With, the method of weighted summation (least-
squares method), when the averages for persons and for
tests have been equalized (a process which is virtually equiva-
lent to eliminating the corresponding general factor), the
factor-loadings obtained by covariating persons are identical
with, the factor-measurements for persons obtained by co-
variating tests multiplied by the standard deviation of the
corresponding factor.

(£) The factor-loadings obtained from a table of co-
variances^ when divided by the standard deviations of the
tests, yield saturation coefficients that will satisfy the
corresponding table of correlations.

(c) The variance of mental traits when directly measur-
able appears to vary closely with their complexity; and,
when not directly measurable, may be treated as equal to
the complete communality, i.e. to the sum of the squares
of the saturations for all factors.

5. The group-factors resulting from the group-factor
method tend to have positive or zero saturations only : the
factors resulting from the general-factor methods (other than
the first factor) have both positive and negative saturations,
and are thus bipolar.

(a) The relations between the two sets of saturation
coefficients can be expressed by a triangular rotation matrix,
containing positive and negative multipliers appropriately
placed. Such a transformation matrix provides the best
means of converting one set of factor-saturations into the
other.

(V) The relations between the two sets of factor-measure-
ments are expressed by the same triangular matrix. This
shows that the general-factor measurements may be regarded
as derived from the group-factor measurements by taking
weighted differences between the latter. Thus the bipolar
factors extracted by the general-factor method are essentially
* difference-factors,' and as such may be intelligibly inter-
preted as they stand.

(c} The group-factor method (with overlapping factors if
necessary) yields a speedier and more precise method of