294	^THE  FACTORS  OF THE MIND
certain traits such as sex). I willingly acknowledge how difficult
it is to obtain a satisfactory set of units, above all for emotional
or temperamental characteristics. Thomson himself goes on to
suggest the possibility of ' some system of natural units ' (p. 293);
and it will be of interest to see how this would work out in actual
practice. At the same time, notwithstanding the admitted short-
comings in the rough methods of assessment that I myself adopted,
I cannot believe that the ' accidental and irrelevant differences'
in the actual variances have produced any great distortion in the
essential nature of the factors. I am even tempted to suggest that a
strict adherence to the principle of reciprocity might itself be made
to indicate, or at least to confirm, what is the most convenient
set of units.
Since the publication of his book, Thomson has generously with-
drawn one point of criticism (cf. [136], p. 76) ; and, quite recently,
has restated his own conclusions. At the close of the Reading
Symposium on Factor-analysis he said : " Although I agree with a
non-rigorous use of the reciprocity principle, I must emphasize that
it only can be found rigorously in a very special sample of people
who are all average in ability, and a very special sample of tests which
are all of average difficulty. Prof. Burt will protest that it will hold
approximately elsewhere ; and I readily agree that it will do so if
these conditions are not too flagrantly broken" ([136], p. 107).
That expresses all that I want. I fully admit that a rigorously
exact equivalence or ^c reciprocity' can only be found under the
conditions specified; it must be remembered, however, that other
writers had previously denied that any equivalence could be found
under any conditions.^1
We shall return to this subject in Part III, where an illustra-
tive example will be fully analysed and discussed.
^1 Superficially, no doubt, the reciprocity principle remains incompatible
with Thomson's sampling theory, at least in its original form. Whereas
other theories reduce the number of factors to a minimum, that theory
implicitly assumes the maximum. Classifying the factors as suggested
above (p. 104), Thomson's ^4most probable factorial pattern' includes
every possible kind of group-factor (cf. [132] fig. 12, p. 44): it invokes
factors entering into any combination of I, 2, ..., and n tests respectively.
Thus the number of factors will be ^nC_1 + ^nC_2 + ... + ^nC_n = 2^n — i in all.
Now, when we apply the same < analysis by overlap' to the correlations
between persons, we obtain an entirely different number, viz. 2^N — I.
Hence the factors, as thus defined, cannot be the same. However, as soon
as we cease to identify each ^£ factor' with a fixed ^e ability,' such incom-
patibilities no longer matter.