472 APPENDIX I

to nearer -63 in the second column. In the fourth column, the first
computation gave -58, the second reduced it to -556, and we guess
that the third or fourth will reduce it still further to (say) -54. And
so for the remaining columns.

(V) yd Approximation (Table FI, c)

5. (a) Multiply each row of the original correlations once again by
these readjusted weights : (-7216 X i-oo = -7216, etc., as before ;
•58 X -75 = '4350 ; etc., etc.). (4) Add each column of products
as before. The largest of the totals (2-1717) yields a slightly closer
approximation to the factor-variance ; and once again all the totals
are to be expressed as ratios of this value. Accordingly—

6. Divide each of the new totals as before by the largest in the
set: (2-1717 -T- 2-1717 = i-ooo; 1-6250 -~ 2-1717 = -7483 ; etc.).
The new ratios agree with the preceding to two decimal places ; and,
to shorten the illustration, we may suppose that they are sufficiently
exact for our purpose.

We now convert the ratios into saturation coefficients. The
requisite calculation is obvious when we remember that the
sums of the squares of the saturations should yield the variance,
which we have already ascertained. The beginner may conveniently
make the calculation in two steps.

7. First normalize the ratios, so that their squares add up to
unity: i.e. (a) square each ratio: (i-oooo2 = i-oooo ; -7483* =
•5599 ; etc.) ; (V) add the squares : (1*0000 + "5599 ~|- . . . +
-4193 = 3-1317). (c) find the square root of this total (^/S'ljl1? =

1-7697) ; (d) divide each unsquared total by this square root :
(i-oooo -r 1-7697 = -5651 ; -5599 -7^1-7697 = -4229 ; etc.) ; (<?)
check the results by squaring and adding the squares, to ensure that
their total is unity : (^Sl2 + -42292 +..«-(- -3659* » i»oooo).

8. (a) Find the square root of the variance : (^2*1717 s= 1-4737).
(4) Multiply each of the quotients obtained in 7 (d) by this square
root: (-5651 x 1-4737 — -8328 ; etc.). The practised calculator
will condense the last two sets of computations into a single step,

and simply multiply by A ...... ^-69348 = -83276,

The products finally obtained (-8328, -6232, etc.) are the satura-
tion coefficients for the first factor. The result should be checked
by finding the sum of the squares of the saturations ('8328* + . . .
+ *539^2 = 2-1717). The value for the * factor-variance ' as thus
determined should be identical with the value already obtained by
summing the first column,