HIERARCHICAL CRITERIA 341

rest. Obviously if the first latent root — i.e. the factor-
variance for the first factor — is equal to the total test-
variance and all the other latent roots are zero, the amount
of separation is a maximum. We may, therefore, assess the
hierarchical tendency of any given matrix by comparing the
figures in its leading diagonal with those in the leading
diagonal of its square or higher power.1

Let /a (— rig in Spearman's notation) denote the satur-
ation of the zth test with the first (or ' general ') factor
(g), and similarly for the other factors. Let ^ = S/2tl
denote the factor-variance for the first factor (i.e. the
largest ' latent root ' of the correlation matrix), and
similar]y for other factors. And let trR denote the c trace '
of the matrix R, that is, the sum of the elements in its
leading diagonal (i.e. of the c self-correlations,' ' test-
variances,' or c communalities,' as they are variously called,
when R is a correlation or covariance matrix.) Then, if
R is hierarchical, trR = S/2tl ; and, on squaring and re-
squaring this hierarchical matrix, we shall have

_

~~ ^ ( Y)' ' ' '

all equal to unity. If, however, R is not hierarchical, any
one of these * trace-ratios ' may still be used to measure its
hierarchical tendency, which we can now define as the speed
with which self-multiplication produces the hierarchical
form. Each of the ratios, as I hope to show, can be given
an easily intelligible interpretation ; and each has its own
special advantages in answering a special form of the funda-
mental question.

Adopting the terminology proposed in a previous paper the three
ratios may be called the criterion of first, second, and fourth
moments respectively.2 Now the use of second moments, as is
there explained, is equivalent in principle to the well-known device
of correlating the columns of correlations. And since this is a more
familiar conception I shall endeavour to elucidate the special merits
of each of these three criteria from that particular standpoint,

1 A more rigorous but technical proof is given in [115], p. 162.

2 [93], P- 28?> footnote I ; [102], p. 178.