DERIVATION OF CHIEF THEORIES 151

if R is a c hierarchy ' in the sense here adopted, i.e. a sym-
metric matrix of rank one, then, as is shown in the textbooks
of matrix algebra, R may be analysed into the product of a
single column- vector f 19 i.e. a matrix consisting of one column
only, post-multiplied by f/, i.e. the same matrix transposed
to form a single row.

Once again, no empirical correlation table is likely to obey the
requirements of a rank-one matrix precisely. But, before we can
decide whether the discrepancies are really indicative of further and
more specialized * group-factors,' we have to discover what is the
best fit obtainable with a single general factor alone. In certain
cases, we have seen, we can regard the coefficient of correlation as
stating a proportionate frequency — i.e. as the ratio of two fre-
quencies, and so analogous to a probability. This suggests a poss-
ible line of treatment.1 Applying the product theorem, we can treat
each expected correlation as the product of two such proportionate
frequencies. To find the latter we may follow the principle adopted
in testing the homogeneity 2 of other double-entry tables : viz.

moment. The converse was formally proved by Garnett ([37], 1916, p. 365 ;
cf. also [78]). His argument proceeded by applying the methods of analytic
geometry, in a way very similar to that later followed by Thurstone. He
concluded that, " were Burt's conditions for a hierarchy satisfied, each of the
n qualities tested would be expressible in terms of two independent factors,
of which one was specific, appearing in that quality alone, while the other
was a single general factor common to all the qualities " ; but if these
narrower conditions are not satisfied, while Spearman's c correlation between
columns ' conditions are satisfied, then " the differences between the test-
measures and a real multiple of an n ~\- ith variable, y, independent of them.
all, can be expressed in the same way." Garnett himself holds that the
conditions for a hierarchy are likely to be satisfied only in very special cases
(when tests affected by group-factors are omitted or pooled). When neither
condition is completely satisfied, he proposes to introduce i independent
variables that will no longer be specific factors ' ; thus finally writing —

= ls . g + . . . + sml . ZJL + Sm2
where g, zlt £2, . . ., and xt are the general, group, and specific factors res-
pectively.

1 This attempt at a logical analysis of the situation was criticized as lacking
in rigour : but, granted certain quantitative assumptions, the argument can
easily be put into a more rigorous mathematical form : (e.g. [93], p. 281).

2 " A classification is homogeneous . . . when the principle of division is
the same for all the sub-classes of any one class " ([25], p. 71) : thus, if we
regard the column of marks obtained by the *th examinee as forming one