SIMPLE AND WEIGHTED SUMMATION METHODS 321

imagine our data plotted as points in a multi-dimensional frequency
diagram : then the orthogonal matrix L will evidently specify the
directions of the principal axes of the frequency ellipsoids — an
interpretation which has suggested what Hotelling calls the
6 geometrical meaning ' of his procedure and Kelley's extension of
the well-known trigonometrical solution of the two-variable correla-
tion problem. Finally, I have myself shown1 that the same equa-
tion is reached if we regard our analysis as an ordinary problem in
multiple correlation, and solve it in the familiar way by the method
of least squares : instead of seeking the best-fitting single line for a
two-dimensional array of points (as we do in finding a suitable
coefficient to express a simple correlation between two variables
only), we now seek the best fitting f-dimensional sub-space for an
^-dimensional array ( r being the rank of the factorial matrix and n
that of the original matrix of test-measurements). This second set
of methods, therefore, consists essentially in the time-honoured
Hauptachsentransformation that constantly crops up in the solution
of so many mathematical and physical " problems of best fit," and
now reappears in slightly varying guises.

(ii) As regards practical computation, the requirement by its
very nature at once suggests some form of progressive approxima-
tion : thus, we might first try to guess the approximate saturations
as nearly as we can ; and, taking the guessed figures as trial multi-
pliers, calculate the saturations resulting from these ; then, if the
results do not tally with the initial weights, we could repeat the
process until they do. The t iterative procedure ' proposed by
Hotelling [79], the successive £ rotation of axes ' proposed by
Kelley [85], and the process of c repeated substitution ' employed
by Rhodes [93], all follow some step-by-step procedure such as this.
Now, my own contention is that if, instead of starting with any
plausible guess (as, for example, Hotelling a appears to do), we start,
as it were, from scratch, putting the weights all equal to unity, and
then mechanically calculate and recalculate the approximate

1 [931 PP- 247 f- ; cf- ab°ve, p. 164.

2 " Round numbers may be chosen at the beginning, and the process will
converge to the correct values anyhow" ([79], p. 431)- I do not suggest
that my mechanical procedure is quicker than Hotelling's, when only two
or three decimal places are required. It is, however, easier for the beginner,
who has no experience to guide his guesses ; and, as explained below, the
mechanical procedure followed by the construction of a geometrical pro-
gression seems speedier and more accurate if highly exact figures, running to
many decimal places, are needed (Miss Dewar and I, for example, used it,
when we desired to show that correlating persons and traits leads to arith-
metically identical figures).

21