GENERAL- AND GROUP-FACTOR METHODS 303

possibly be assigned to c a factor which improves certain test-
performances when it is not merely absent, but actually negative '
or to ' an ability whose possession is a detriment to performance ' ?
(Cf. [82], pp. 99, [84], pp. 161, 165-6, [122], p. 71). The procedure
recommended,1 therefore, would doubtless be to c rotate the axes '
until a simpler and more intelligible factor-pattern was secured.
For this purpose the plan usually followed2 is to plot graphs for each
pair of factors, fit fresh axes at right angles to one another by eye,
and measure the angles between them and the original axes with a
protractor. But so rough a device must obviously yield somewhat
arbitrary and inexact results.

The relation between the results of the group-factor
method and those of the general-factor method can be
expressed by a simple transformation matrix. Elsewhere I
have described how such a matrix may be computed, and
have shown how it yields a direct procedure—far more
exact than the usual graphical devices—for rotating axes to
abolish the negative saturations [i 16], As calculated for the
present figures, it is shown in Table IIIo (T). It is evidently
a 5 X 5 orthogonal matrix with one of its rows deleted :
hence the transformation matrix for the inverse operation
(which we should denote by T7"1, were T non-singular) is
simply the transpose of the previous, namely T' (Table
IIIc).

In order to illustrate the derivation and structure of rotation
matrices such as T, let us suppose that (with Thurstone and Alex-
ander) we have started by analysing our correlation table according
to the c general-factor method/ and now desire to abolish the nega-
tive saturations and to maximize the zero saturations in the factorial
matrix so obtained, namely Fb. In theory the problem is familiar
enough : it is evidently that of obtaining for Fb, by a series of
elementary transformations, an ' equivalent matrix/ fulfilling the
specified requirements. The simplest procedure is to keep operating
on the columns (or rows) of the given matrix (and on the results of
our operations) by the ordinary methods used in the reduction of
determinants, until we reach a pattern of the type desired. If the

1 Employed or suggested by Thurstone, Thomson, Alexander, Guilford,
Stephenson.

2 " The graphical method of rotating in one plane at a time is still probably
the best. . . . But the graphical method is not ideal" (Thurstone, 1938,
[122], pp. 72-3).