NATURE OF THE FACTORIAL TECHNIQUE 79

pins. With n traits instead of 3, the vectorial pattern will
be specified by co-ordinates in n dimensions : but, if the
student has trouble over picturing patterns in ^-dimensional
space, I suggest he translates the n-dimensional configura-
tions into flat zigzag c profiles' by treating the co-ordinates
as ordinates.

Factors as Axes of Co-ordinates.—When, however, we
try to enlarge not only the list of traits but also the sample
of persons, when, that is to say, we have to deal simul-
taneously, not with one or two individuals only but with a
very large number, and possibly with a number of different
types to suit the different groups, then some form of
//-dimensional representation, or at least the language of
7z-dimensional representation, becomes inevitable. We
start with N persons tested and measured for n correlated
traits ; and we desire to convert this empirical mark-sheet
into a more convenient form by describing the same N
persons in terms of r uncorrelated factors. How is the
pattern of weights to be deduced ?

When the problem is put in this generalized fashion, the
mathematical arguments take on a somewhat formidable
aspect. For the elementary student the simplest mode of
exposition is to outline the proof for two or three variables
only ; and then show that the number of variables is really
irrelevant to the argument so that the proof can. be general-
ized for any number. But if matrix algebra is used, the
proofs for variable matrices are almost as simple as the
proofs for variable scalars.1 This was the procedure
adopted in my early Notes [93]; and algebraic proofs need
not be repeated here. In general the transformations
employed follow the methods regularly used in elementary
geometry for translating measurements obtained in terms
of one set of co-ordinates, presumably a provisional or
casual set, into terms of a second set, chosen so as to be

1 Nearly all Spearman's proofs relating to a single factor, for example, can
be generalized in this way. It is curious that Thurstone, after his lucid
introductory exposition of matrix algebra, uses it so little in his subsequent
proofs, relying instead almost entirely on the old summation notation ;
(cf., for example, the simplicity of the matrix proof of the formula for
* appraisal of abilities' with that given by him in [84], chap. x).