266	THE FACTORS OF THE MIND
saturations by the matrix of correlations between the several
factors. When the factors are orthogonal, the latter
becomes a unit matrix. Hence (ii) with orthogonal factors,
as we have seen, the correlations between factors and tests
are given directly by the factor-saturations ([101], p. 88).
An analysis which yields oblique factors may be regarded
as an incomplete or partial analysis, since all correlation
has not been eliminated; and the factors so obtained may
be regarded as c partly analysed' or c mixed factors.' The
two modes of analysis correspond to the two kinds of
* selective operators' which the physicist has recognized :
first * fractional' operators, which effect a partial analysis
into constituents that are not pure ; secondly, £ spectral'
operators, which effect an analysis of a mixed aggregate
into pure constituents. The difference is usually defined
by saying that fractional operators, unlike spectral operators,
are not idempotent ([115]? p. 159)-
In theoretical inquiries the advantage of working with indepen-
dent factors will be sufficiently obvious from analogies with partial
differentiation and partial correlation. It is, however, part of an
ultimate logical requirement :x all reasoning about complex
variations becomes, not only simpler, but more rigorous, if we can
reduce it to terms of concepts of which any one can be taken to
vary, while the others remain constant. In factor-analysis we are
enabled to reduce the number of eifective elements in the covariance
matrix from \ n (n + i) to n; and the terms eliminated (the co-
variances as distinct from the variances) are precisely those which
are most difficult to manipulate algebraically, to compute" arith-
metically, and to estimate statistically. In short, with independent
variables, we can treat the variances as simply additive.
In practical applications such orthogonal factors may not always
be necessary or even desirable. Eventually, perhaps, we may be
able to identify certain of our so-called factors with well-marked
physiological agencies or mechanisms, e.g. the secretions of certain
glands, or * unit characters ' in the genetic sense : but the activities
of the various glands will almost certainly show some low degree of
correlation ; and some of the unit characters will be, not indepen-
dent, but linked. Meanwhile, if we regard our hypothetical factors
1 On the logical importance of independence in its various forms—notional,
connectional, implicational, and the like—see Johnson, Logic, pp. 54 f.,
108 f.? 21 o, f,