52	THE  FACTORS OF THE  MIND
c/t^ — 1/3. We can then imitate A's results by averaging the true
order with a second order uncorrelated with it * ; and we can imitate
B's results by averaging the true order with two other orders
uncorrelated with it and with each other. According to the simplest
interpretation of correlation, if t_a is the total number of independent
elements determining a variable #, and c the number of elements
common both to it and to another variable g consisting wholly of
those common elements, then it can easily be shown^2 that
_lg = A/ — ;  similarly, rb_s = A/ - ;  and
^   ^«	^
f	n   /~
,   		^rr^r .  f^of i"o    *•   .   	    A   /   -~-        A   /	'   /'
ib —    /	• LJ.JL<tL io, fab —    \/         *     V/         —'    ^8 '
\ ^a. ti)	$0,	*b
^1	A worked illustration will be found in my Note on Correlation as Applied
to Mental Zesting, Board of Education Report [48], pp. 188-9.
^2	Cf. Bowley, loc. cit., 1904, p. 355.   These ^f ratio-formulx * are based on
Bravais's method of deducing the correlation rather than Pearson's (^c Analyse
mathematique sur les probabilites de situation d'un point,' A cad. Sci. : mem.
sav._9 ii^6me ser., t.IX, 1846, pp. 255 f., eq. 28).    Pearson's deduction of the
product-moment  formula  proceeds  by taking the product-sums  of the
empirical measurements (obtaining r = mm ') ;  Bravais's original deduction
proceeded by taking the product-sums of the independent elements of which
those measurements were supposed to be composed (obtaining r — jf').
Bravais's mode of approaching the subject, though heavily criticized in the
past (cf. [47], p. 152), is full of suggestive points for the factorial, which
the other approach has led us to neglect.
The idea that correlation may be regarded as expressing the proportion
of common elements has appealed especially to those interested in heredity.
This was the origin of the celebrated experiments of Darbishire and Weldon
on artificial correlation obtained by throwing dice [14] ; and it still finds
favour as a principle of teaching and interpretation (cf. Snedecor, he. cit*
sup., pp. 128, 130 : " Roughly this is the interpretation of the father-son
correlation in stature, which is not far from -J : an average of some 50 per
cent, of the genes are common to father and son."). Thomson adopts a
similar standpoint in his * sampling theory,' which he introduces with an
ingenious extension of Weldon's experiments and formulae ([132], p. n).
Here, however, it seems important to ask whether Thomson's elements
(^c neurone arcs ') are as independent as the biologist's elements (^c genes ') :
for the neurones belonging to the same individual have all developed from
the same material. This, however, is a point we must take up later on.
Meanwhile, it may be noted that, if mental factors or * abilities ' are
regarded as equivalent to forces, then the product-theorem is the equivalent
of the familiar formula for the composition of two forces. This analogy,
however, is also full of unsafe suggestions, unless its precise grounds are made
clear (see below, p. 91), Indeed, I fancy it might be legitimately maintained