308 THE FACTORS OF THE MIND

manifest) the transformation itself is of such a nature that
the initial factor-pattern obtained before transformation
must also bear a related interpretation. This can be
confirmed by considering the relations between the two
sets of factor-measurements.

Let us suppose that we have reached, directly by the group-
factor method or through an appropriate transformation, a simple
and satisfactory factor-pattern. The five factors themselves we shall
endeavour to identify from the tests or traits we have been correlat-
ing ; let us call them (to keep the illustration concrete) (i) general
intelligence, (ii) verbal, (iii) manual, (iv) arithmetical, and (v)
artistic capacity, respectively. Let us further suppose that we have
calculated the hypothetical measurements of our tested population
for these five group-factors,1 and desire to deduce their measurements
for the equivalent general factors. Which of the group-factors
must we take, and in what proportions must we combine them, in
order to obtain estimates in terms of a given general factor ? Call
the first set of factor measurements Pa (i.e. c population matrix for
group-factors as obtained by method a') and the second Pb (i.e.
4 population matrix for general factors as obtained by method b ')•
We require a transformation matrix T, such that TPa = Pb.
Curiously enough, T as thus defined turns out to be identical with
the rotation matrix T already obtained.2 In other words, T not
only converts Fb into a c primary ' structure Fa ; it also tells us how
to weight and add or subtract the group-factor measurements in
Pa so as to obtain the general-factor measurements in Pb.

It thus becomes evident that the rotation matrix already
obtained not only enables us to transform the one set of
saturation coefficients into the other, it also describes the
relation between the first set of factor-measurements and the

1 The first of the five would ordinarily be called a c general factor/ or
rather * the general factor *; but, to save circumlocution, I may perhaps
be permitted to use the phrase c group-factors * to mean factors obtained by
the * group-factor method of analysis.'

2 I have given a formal proof elsewhere ([116], pp. 339-75); but the
following perhaps makes the relations somewhat clearer. Adopting the
same notation as before and using the weighting equation xxxvii ([93],
p. 299), we have:

pa = F'a £-1 M = (Fb TY R-iM=T'F'b R-* M=TPb.
But TT' = /. Hence Pb = TPa.