VARIOUS USES OF FACTORS        53
as may be roughly verified by direct calculation from the averaged
orders.1
The fact that direct prediction is safer than indirect is
quickly realized by any investigator who engages in educa-
tional and vocational guidance and is able to follow up his
cases over an appreciable period of time. In earlier writings
I have sought to stress this caution more than once. Never-
theless, these reservations have often been overlooked both
by those who rely on psychometric methods and by those
who criticize them. Accordingly, it may be advisable to
consider in more concrete detail the two commonest cases
in which attempts are made to base a practical prediction on
that the compounding of forces is itself a compounding of logical implications,
not of actual causal entities. In psychology at all events, once the student
has grasped the notion in the concrete, he should, I hold, look upon it as
a principle of reasoning rather than as a principle of causation.
1 The formal proof on which I have usually relied proceeds by writing
mki = fikg . g± -|~ sm J the product theorem follows at once on correlating
mai and m\^ ([93], p- 281, eq. xix). If we express the correlations as cosines
and rewrite this initial equation m^ = cos 0A . gt + sin 6k . j#, the same
argument brings out the analogy between what I may callc factor-synthesis '
and the composition of forces, and, conversely, the analogy between factor-
analysis and the resolution of forces.
The shortest proof is that obtained from Yule's formula for partial
correlation ([25], p. 238, eq. 12), by putting the residual correlation r12.g = o ;
but this formula really assumes the product-theorem to prove the residual
correlation instead of vice versa, and the deduction does not make the
theorem clearer to the student who has not followed Yule's somewhat lengthy
demonstration. For the non-mathematical student the argument (with
the illustrative exercise suggested in the text) seems to be the clearest. I
relied upon it in my earliest paper because I then assumed that the corre-
lation of' a test with the common factor might be plausibly supposed to
depend on and increase with its complexity. The instance there given is worth
recalling because it illustrates the complexity theory of the general factor,
which still seems to me to contain an important element of truth.
To quote my original example ([16], p. 160), suppose we have a series of
sensori-motor functions, each of varying degrees of complication, yet all
essentially manifestations of one common process, say motor co-ordination
(X] : for instance, we may imagine each to consist of a different number of
elementary sensori-motor reactions, added or otherwise combined, so that
the most complicated test, A^ is equivalent to, say, a dozen determinations,
and the most simple, J?, to only three. Then, applying the ratio-formula,
we can find a measure of the influence of X on A and B by computing its
hypothetical correlation with each: and similarly, we can compute the