DERIVATION OF CHIEF THEORIES 149

columns) are proportional to one another, can always be
expressed in terms of a single c factor,' that is, as the product
of a single column of * factor loadings' or (saturation
coefficients? post-multiplied by a row of precisely the same
coefficients." x Such a covariance table of rank one it will
be convenient to designate by the brief and familiar name,
a ' hierarchy.'2

1 In more accurate and technical language : " Any symmetric n X n
matrix of rank one can be expressed as the product of a one-rowed matrix
(or ' vector ') pre-multiplied by its transpose, the elements in the one-rowed
matrix being the square roots of the diagonal elements of the symmetric
matrix " ; or, in symbolic form, if r^ denotes the element in the jth row of
the ;th column in the symmetric matrix R, i and j standing for any row and
column, then r# = A/^ X AAjy- I*1 dealing with an actual correlation table
(as distinct from a covariance table) the chief difficulty arises from the
absence of values corresponding to r#, r#, which have to be indirectly com-
puted (see below, footnote I, p. 152). The examples given in Appendix I,
Tables I and III, will make the theorem quite clear; a simple formal proof will
be found in Cullis, Matrices and DeUrminoids, II, 1918, equation A, p. 134. I
may add that it seems convenient to keep the term " saturation coefficients "
for the elements of a factorial vector obtained from a correlation table, and
use the newer term " factor loadings " for the elements from a covariance table.

2 It will be noted that the above conception of a * hierarchy' is in some
respects narrower, and in others broader, than that adopted by many
writers : one or two of the tables given as examples of a perfect hierarchy by
Spearman and his co-workers would not be hierarchies by my definition (e.g.
Brit. J, Psych., 1916, VIII, p. 175 ; Psychology Down the Age$> II, 1938,
p. 274) ; on the other hand, they, I gather, would not accept my bipolar
matrices of rank one as hierarchical, nor yet the covariance tables which
are not correlation tables.

The original and more usual definition of the hierarchy simply required that
the correlations could be placed in " an order such that each is greater than
any to the right of it in the same row, or below it in the same column " [12]
With the small tables previously employed, this could be satisfactorily judged
by eye. With larger tables, containing as many as 156 coefficients, and with
group-factors tending to disturb the order, a more precise procedure seemed
essential [17], Accordingly, assuming that a perfect hierarchy would obey the
product-theorem, the test proposed was that all the residuals should be cal-
culated and shown to be attributable to chance—admittedly a cumbersome
procedure. Later Prof. Spearman proposed to test the orders in the rows by
correlating the rows (or columns). By this criterion, a table, like Carey's, in
which the correlations diminish arithmetically would be accepted as a
hierarchy: whereas with my procedure it would not. Spearman's final
criterion, however—that the * tetrad-differences' shall all be zero—is vir-
tually identical with the criterion for a matrix of rank one, except that he
further implies that all the correlations must also be positive.