DERIVATION OF CHIEF THEORIES 165

On this * canonical expansion of the correlation matrix '
all factorial methods, in my view at any rate, ultimately rest.
In virtue of this theorem every correlation matrix can be
expanded as a sum of weighted hierarchies* It thus plays
much the same part in the factorial analysis of product-
moment functions as Fourier's theorem plays in the har-
monic analysis of periodic functions, where, it will be remem-
bered, any such function can be expressed as a sum of weighted
sines. Moreover, in psychological work, as we shall pre-
sently discover, no matter how large the correlation matrix
may be, the factor-variances successively obtained from it,
z'l? ^2? ^35 • • •? nearly always form a series that converges
very rapidly. This, indeed, is the reason for the common
statements that in psychology the correlation matrix can
always be accounted for by a single general factor only, or
(as more cautious writers put it) that the correlation matrix
always has a rank of one or nearly one. Neither suggestion
is quite accurate. But, just as with our ordinary number
system we can express any number, rational or irrational,
as the sum of a converging series (e.g. —

3-141 . . . = 3 + ^ + T^ + T^ +...)>

and then use the first two or three terms only as a round
approximation, so here : having expressed the correlation
matrix as the sum of rapidly diminishing terms, we need
keep only the first two or three terms, 'discarding all that
are within the margin of error ; and these first two or three
will give a close approximation to the original matrix.
Finally, we may note, it is this canonical expansion that
reveals the striking parallel between the * factor-analysis '
of the psychologist and the so-called c spectral analysis ' of
the quantum physicist, which happens to turn on an almost
identical equation (see [115], p. 160).

This particular mode of analysing the matrix of correla-
tions leads to an equally simple mode of analysing the initial
matrix of measurements, which, after all, is the table that
primarily calls for analysis ; for we may now write ([101],
p. 75, [102], p. 177)—