243 THEORY OF STATISTICS. 
For we have 
(xy. e103... n) = 22) (2) — Ty 45 Sci n)=N(oF — 034 nin) 
and also 
(ef . . . n) = 22) — Fy 95 Joti n)?=N(a}- on... .n 
whence the correlation between Zande... . ,is 
04 ! 
v.e. the value of By, n given by (13). The value of RB is 
accordingly a useful datum as indicating how closely x, can 
be expressed in terms of a linear function of Fo Fini oi so « &py ANA 
the values of the regressions may be regarded as determined 
by the condition that & shall be a maximum. Its value is 
essentially positive as the product-sum 2(xy.e95 . . . . 5) Is positive. 
£ may be termed a coefficient of (n—1)-fold (or double, triple, 
etc.) correlation; for n variables there are » such correlations, 
but in the limiting case of two variables the two are identical. 
The value may be readily calculated, either from Tio, .. 800 
o, or directly from the equation 
1 - Bios... w= (1 =18)1 - 7352) ~ 72s). . . (1 = Puss... (ney). (15) 
It is obvious from this equation that since every bracket on 
the right is not greater than unity, 
LB... Pl =7% 
Hence Ry; , . . . , cannot be numerically less than r,,. For the 
same reason, rewriting (15) in every possible form, Ba 
cannot be numerically less than ry, Tig + «+ « Tip 2.6. ANY ONE 
of the possible constituent coefficients of order zero. Further, 
for similar reasons, Ryn; » cannot be numerically less than 
any possible constituent coeflicient of any higher order. That 
is to say, Ry... , is not numerically less than the greatest 
of all the possible constituent coefficients, and is usually, though 
not always, markedly greater. Thus in Example i, Buy 
(the coeflicient of double correlation between pauperism on 
the one hand, out-relief and labourers’ earnings on the other) 
is 0-839, and the numerically greatest of the possible constituent 
coeflicients is 1,,= —0'73. Again, in Example iji., By, is 
0:626, and the numerically greatest of the possible constituent 
coefficients is ry, , = + 0-573. 
The student should notice that R is necessarily positive. 
Further, even if all the variables Xp Xp... . X, were strictly 
uncorrelated in the original universe as a whole, we should expect 
"120 "13.90 "14.93 €tc., to exhibit values (whether positive or negative) 
By