480 APPENDIX I

8. Divide the curtailed totals of the columns by the appropriate
divisor (1-89 4- '21 = -90, etc.).
The result will be the saturation coefficients for the first factor.

(ii) Second Factor

Formula : Residual correlation, rfa^ = ?a& — rag ?V

z/*

Saturation coefficient, r'as = ••--, * - f -

A/ i^f y*

V j *

1. Multiply each of the saturation coefficients by each of the
others, thus obtaining a perfect hierarchy of theoretical correlations
attributable to - the first or general factor (-90 X "80 = ^72,
•90 x -70 = -63, etc.). The multiplication proceeds a& before, and
its results are given in the lower of the paired rows in Table IX.

2. Enter the observed figures above them, and subtract the
theoretical figures from the observed, as shown in Table IX
(*75 — *72 = *°3> "65 — '63 = -02, etc.). When (as here) group-
factors are discontinuous and do not overlap, the residuals thus
obtained will be approximately zero in each cell of the rectangular
blocks governed by the general factor alone ; but in the square
blocks along the leading diagonal, where the effects of a group-
factor are superimposed on those of the general factor, there will be
positive residuals due to the operation of the former.

3. Take each square block of residuals in turn, treat it as a separ-
ate hierarchy, insert estimated values for the diagonal coefficients,
and analyse the whole by the method described in Section IA.
In Table X the central block of residuals from Table IX is
factorized as an example.

The four sets of saturation coefficients obtained by this twofold
procedure are set out in Table XL It will be seen that they account
perfectly for the ' observed ' correlations. To render the illustration
more definite these correlations were artificially constructed at the
outset from these very figures. And it will be noted that the group-