50	THE FACTORS OF THE MIND
central theorem in all factorial work. I have stated it as
follows: " If test I is correlated with the only common
factor g to the extent of r_ls and test 2 is correlated with
the same factor to the extent of r_2g, then test I and test 2
will be correlated amongst themselves to the extent of the
product of those two correlations, namely,, r_12 = r_ls, r_2g: " ^l
more briefly, the correlation between two variables is the
product of their correlations with the only common factor.
When we apply this theorem for purposes of prediction
and the like, it is important to remember that we are dealing
only with an estimation of r_12 and the corresponding
regression coefficient. No doubt, in theory we can show
the first fraction will be unity. Similarly if the second premiss is universal.
It will then follow that the third fraction must also be unity, i.e. that all
the children who succeed in the test will also gain scholarships, as the
syllogism concludes.
(2) If (as we assume in paragraph 2) only some of the children who succeed
in the test prove to be really intelligent, then we can insert the actual
number, e.g. the average as deduced from our sample ; and the first fraction
will be less than unity. Similarly for the second premiss. Taking the
probability to be the ratio of the two frequencies (or, more accurately, the
limit approached by this ratio as the number in the denominator is increased
indefinitely) we arrive at the multiplicative theorem stated in para. 2, p. 47,
If the argument is kept in terms of frequencies, it is equivalent to that used
in the association of attributes. Thus, employing Yule's notation, let A
= number who succeed in the intelligence test, JB = number who gain
scholarships, C = number who are intelligent, AC = number who are
intelligent among those who succeed in the test, and so on. Then, if intelli-
gence is the only common factor, there will be no partial association between
A and B within class C ; and bab c = °- Accordingly, by Yule's formula,
(ABC] (AC] (BC) , , .. .' _r .	_r .	, _s _T
(2o   ^=W'W	^P> ^49;	^PP*
passing, it should be noted that the < criterion for independence' within
class C is really a criterion for showing that the factor responsible for the
classification into C and not-C is the only common factor ; we shall recur
to this below (p. 147).
(3) Finally, with a slightly different line of reasoning, for " average
number of (children who are also intelligent/' etc.) we may substitute
" average deviation of (the same children, in intelligence)," with similar
substitutions in the other fractions. We then reach the last form of the
product theorem as stated in paragraph 3.
^1 The Measurement of Mental Capacities•, pp. 11-12: ci also Brit* J, Psych.,
1909, III, pp. 159-60; L.C.C. Report, 1917, p, 53, equation (ii). The
application of the theorem is fully illustrated in each of these publications.