WORKING METHODS FOR COMPUTERS 473

(iii) Direct Verification (Saturation Coefficients as Weights).—As
with the perfect hierarchy (cf. Table II above) so with any table,
the exact or the approximate values for the saturation coefficients
may themselves be used as weights. With the appropriate divisor
the method of weighted summation should then lead back to the
saturations originally employed. This more direct method is
especially suitable for the final verification (and minor corrections)
of the figures obtained by one of the methods of successive approxi-
mation ; it may also be used, without such preliminaries, when the
table is nearly hierarchical or when only a rough and rapid approxi-
mation is required. In the latter case it affords a useful check on
figures obtained by simple summation.

Here let us take the figures just reached in Table VI, c, as our
weights. We proceed as for Table II above, and follow the working
instructions explained on pp. 451-4. For the present set of
correlations the working is shown in Table VI, d. The product sums,
it will be remembered, have simply to be divided by the variance,
which in turn is estimated by dividing the total of the product-sums
(7-6741) by the total of the weights (3-536). The saturation
coefficients thus obtained are, it will be seen, virtually the same as
those just used for weighting.1 The simple divisor here employed
does not eliminate from the product-sums any constant factor which
may have increased or decreased all the weights in equal proportion.
Usually, therefore, the variance as thus estimated must be finally
checked by adding the squares of the saturations. In the present
example an almost complete agreement has already been secured by
the final calculations in Table VI, c. Should the two values not agree
precisely, we should have to divide by the square root of their ratio
as before (cf, p. 454)-

Furthtr Factors.—To obtain saturation coefficients for the second
and subsequent factors (if required) we follow the general procedure
already explained. A theoretical hierarchy is constructed by
multiplying the first factor-saturations by each other; the residuals
are then analysed by one of the methods of weighted summation just
described.

The complete set of saturation coefficients obtained by weighted
summation is shown in. the last three columns of Table VIL It
will be seen that the sums of the squares of the saturation coefficients
for each test are precisely the same as those obtained by the simple
summation method, being in each case those required to complete

* I have carried the calculations to one more decimal place in order to facilitate
the demonstration of certain equalities in Table VII. This has yielded a slight
modification in the figures for the third and last tests: and incidentally the modifi-
cation secures a perfect agreement in the two estimates for the variance.