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 (— r_ig 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/^2_tl
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/^2_tl ; 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.