XVL.—NORMAL CORRELATION, 221
isotropic. The student should form one or two other condensations
of the original table to 3- x 3- or 4- x 4-fold form : he will probably
find them either isotropic, or diverging so slightly from isotropy
that an alteration of the frequencies, well within the margin of
possible fluctuations of sampling, will render the distribution
isotropic.
14. Before concluding this chapter we may note briefly some
of the principal properties of the normal distribution of frequency
for any number of variables, referring the student for proofs to
the original memoirs. Denoting the frequency of the combination
of deviations x, @, «,, . . . , x, by Yi2 . ... n We must have
in the notation of Chapter XII. if the uncorrelated deviations zy,
Typ» ¥3.19: ete. be completely independent (cf. § 3 of the present
chapter),
Viz... n=Yis. .. ne CHE 2 ta) (12)
where
z. x = “a1 Pitiy nS Tai... ey
Ht... @) AT AL ore (13)
gad ¥12 eee? (27)"*6,04103 19 w ele Cet FRYE ae inl) (14)
The expression (13) for the exponent ¢ may be reduced to a
general form corresponding to that given for two variables, viz.—
Be vn ua... .n Hk rE (n—1) (15)
: Lp —1%y,
2Ti2s...m Cis... A028. ns Lr nti... 2 Tn-1)1... n-9nOnl... (a=1
Several important results may be deduced directly from the form
(13) for the exponent. Clearly this might have been written in
a great variety of ways, commencing with any deviation of the
first order, allotting any primary subscript to the second deviation
(except the subscript of the first), and so on, just-as in § 3 we
arrived at precisely the same final form for the exponent whether
we started with the two deviations xy and x, or with x, and xy oe
Our assumption, then, that the deviations xy, Ty, ¥g49 etc. are
normally distributed amounts to the assumption that all devia-
tions of any order and with any suffixes are normally distributed,
i.e. in the general normal distribution Jor n variables every array
of every order is a normal distribution. It will also follow, gen-
eralising the deduction of § 6, that any linear function of x, X,
+ + + . %, is normally distributed. Further, if in (13) any fixed
af
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