3io THE FACTORS OF THE MIND

were negative, and must first be converted into positive or null
classes.

The transformation matrix that I have described pro-
vides the link between the results of factor-analyses by the
general-factor method and the group-factor method
respectively. The matrix is triangular. In practice it
tends always to show the following features—each plainly
exemplified by the ideal scheme of Table HID : (i) diminish-
ing positive figures in the top row ; (ii) diminishing positive
figures in the diagonal; (iii) zeros below the leading
diagonal; (iv) small but increasing negative figures above
the leading diagonal (except for the top row) ; and—
though this is less uniformly present—(v) figures in all
rows except the first adding up to zero. Its general effect
will be evident from a consideration of these character-
istics. And if we study the process of post-multiplying Fa
(Table III A) by the inverse, namely T (Table IIIc), we
shall at once understand how the several * general factors '
might have been derived from the ' group-factors/ instead
of vice versa. Each column of 7" in turn (excluding the
first) takes one of the group-factors, and gives to this factor
a high positive weight, and to those factors that succeed
it low negative weights, the total of which will approxi-
mately balance the positive. The result is that the satura-
tion coefficients of these succeeding factors are subtracted
from those of the preceding factor. It follows that, when
applied to a population matrix obtained by the group-
factor method, this inverse transformation must in effect
substitute for the positive group-factors a series of bipolar
difference-factors. Hence the real purpose of the subsequent
rotation (upon which Thurstone insists) is to convert these
difference-factors back into the positive factors from which
they were implicitly derived.

The factorial and transformation matrices on which my
argument has here been based are admittedly schematic.
But, once the types have been recognized in their simplicity,
matrices showing the essential characteristics of FM Fb,
and T will be easily discerned in published investigations;
and, by the use of matrix notation, the arithmetical example