WORKING METHODS FOR COMPUTERS 465

negative rectangles arranged symmetrically about the leading
diagonal, as described on pp. 456-8, step 2.1 Follow the suggestions
therefore for marshalling the coefficients in this order. If the table
of observed correlations was successfully rearranged at the very
outset (step 2, p. 461), the amount of further rearrangement will
probably not be very great. The essential result will be two (pos-
sibly more) square blocks of positive coefficients astride the main
diagonal, and rectangular blocks of negative correlations in the
N.-E. and S.-W. quarters (as shown by the residuals in Table V).

11. For the arithmetical check described in step 9, we retained the
calculated residuals in the leading diagonal. But these were the
result of the first rough estimates in step 3 : they may even be
negative. Discard them ; and substitute estimates for the self-
correlations or residual £ variances/ positive throughout, to fit the
general pattern.

12. Assign to each row its appropriate weight, + i or — i. For
this purpose first decide on the block of (mainly) negative residuals,
attributable to the multiplication of negative and positive satura-
tions. Owing to the symmetry of the table, this negative block will
appear twice. Its longer horizontal rows (in the lower quarter)
will usually receive a weight of — I ; its shorter horizontal rows (in
the upper quarter) a weight of + I (cf. p. 457).

13. Multiply each row by the weighting-sign thus prefixed, i.e.
where the weight is negative, reverse the signs throughout the whole
row from left to right of the table.

14. Using the new signs, add each column to find its algebraic
total. See that the sign of each total confirms the sign already
employed to weight the corresponding row.

15. Find the numerical or absolute grand total of these column-
totals : i,e. treat them as all positive, and add them up.

16. Find the square root of this grand total.

17. Divide the total of each column (with its original sign,
positive or negative as the case may be) by the square root. The
quotients should yield the saturation coefficients for the second
factor. As a check on the working, note that their total should be
approximately zero : (exactly zero, as in Table XI below, had we not
modified the diagonal residuals).

(iii) Remaining Factors

18. To obtain saturation coefficients for the third and other
factors, if necessary, we continue exactly as before, calculating the

1 If there are more than one residual factor, these' grand quarters ' will be * counter-
quartered ' j hence only an approximate hierarchy will be attainable at this stage.

3°