CLASSIFICATION OF METHODS 257

factorists, so far as they share a common procedure, would
seem to suggest the following formulation.

(i) Components and Factors Generally. — Let there be n
observable variables measured for N individual cases.
The n X N empirical measurements so obtained — {ml9
mz, . . ., mn} say — will in general be correlated with one
another in a more or less arbitrary fashion. We can,
however,, assume that it is always possible to express them
exactly or approximately in terms of r new hypothetical
variables, {p19 p2, . . . , pr] say, (r=ri)9 which will be
related to each other in some simpler, specifiable way : so
that we can regard the empirical set of measurements as
functions of the latter, and write

Then these r hypothetical variables {pl9 p^ . . ., pr} may
be called * factors,' or more accurately * components/ of
the observed measurements ; and any particular system of
components will be defined by the system of functions /,-,
or the inverse of that system if it has one.

Such a transformation can be effected in an infinity of
ways ; and in carrying it out, it is, as a rule, implied that
the new variables will be so chosen that they can serve as
reference values. Thus, (i) usually the specifiable relations
between them will be the simplest possible, namely, those of
independence or non-correlation ; (ii) usually, too, their
number will be the fewest possible, or at any rate they will
be fewer1 than the original n variables, if only because

1 There are many .important exceptions to this rule, which, however,
would seem to be apparent exceptions only. Thus, Garnett ([37], 1919,
p. 346) takes r^n, envisaging one * general factor,' n specific factors, and a
varying but presumably small number of group-factors. This is virtually
the view of Thurstone (whose proofs resemble Garnett's in many ways)
except that he would not specify a general factor as distinct from the other
common factors, and always insists that the number of common factors must
be less than n. Garnett also mentions the possibility that the one general
factor (and the other factors, if desired) may be so transformed that r may
be any number whatever : this is presumably to allow for Thomson's sampling
theory, where the number of f elements ' is assumed to be indefinitely large
and the number of c factors ' a maximum (viz. 2*"1 : see below, p. 294, foot-
note). Spearman, on the other hand, always takes r = n -f- I. Hotelling

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