268	THE  FACTORS OF THE MIND
a mathematical standpoint such modes of analysis appear partial or
incomplete, like a fractional distillation that only partly separates
the various ingredients.1
Methods Available.—The various methods proposed need
not be described afresh. Formal descriptions of the main
types (taken from roneo'd notes compiled originally for
the use of my own research-students and based on early
articles by the chief authors concerned) were included in
my Memorandum [41]; but the systematic expositions
since published by Thurstone [84], Kelley [85], and Hol-
zinger [106], all of them remarkably lucid and suggestive,
render my previous account of their principles not only
1 A clear and systematic account of the relations between the two types
of factors has been recently given by Holzinger and Harman (J. Educ. Psych.,
1937, XXVIII, pp. 321-45). More recently still, in his contribution to the
discussion on t Factor Analysis' Stephenson has developed the same
distinction in a suggestive way ([137], pp. 100-2). Here, and in a per-
sonal communication, he criticizes my own account in two respects
that deserve a fuller reply than was possible at the Symposium ([137],
pp. 92). In the first place, though he emphasizes the distinction between
' concrete,' * functional,' or e unanalysed abilities' and such abstract or
hypothetical factors as Spearman's * general factor ' (g) and my ' verbal
factor * (v\ he rejects " the supposed * independent factors * as gratuitous
assumptions." Factors like g and v he now prefers to call' fractional factors,'
to distinguish them from the * non-fractional factors' that play a central
part in his theory of Q-technique ([138], p. 272). Yet on a later page he
explains that these 'fractional factors' are uncorrelated, and are obtained by
analysing the concrete or ' non-fractional abilities.' Surely in that case they
are identical with what Alexander and I understand byt independent factors,'
so that after all these latter cannot be f gratuitous assumptions.'
Secondly, he adopts the algebraic argument from the article of mine just
quoted ([115], pp. 157-8) ; but thinks it should be applied, not, as I applied it,
to show that the factors complying with it cannot be further analysed, but to
show what conditions the * unanalysed abilities' must fulfil. This would
make his unanalysed abilities or ' non-fractional factors' identical with
Holzinger's original fi bifactors,5 i.e. bifid factors ([106], p. 6): for in both
cases the saturation coefficients for the two * pure ' factors are assumed to be
proportional. In an empirical matrix, however, such a precise proportion-
ality is so improbable that Holzinger has now given up the conception
([107], p. 53). In the fuller manuscript version of his paper Stephenson
points out that his non-fractional factors are analogous to the factors derived
by Tryon with the aid of * cluster-analysis '; and in his more recent work
(particularly on types with ' Q-technique ') he deliberately avoids a * general
factor,' so that his non-fractional factors are, as it were, group-factors