<7)

��boas] the cephalic index 457

We call

��I 'lap — '123

r 1 3p = ^133

��f 1 3 ^ 1 p |) J

1 — >- 3P r» s = r » 3 i^

� � �. r » 3 ~~HLP. r P 3

1 r 3 p r P 3

� � �p "T~ f iyi . . . . p 7* 3 2 p 1 •

�• 1 f I (p- 1)2.

�. p ^"(p-i)ap

�P^*23P ~| ^" 132 . . . . p 1 •

�• "1 *"i(p-i)a .

�• P r (P-03P

��v ^i(p-i) p = ^"123 . . p ^2tp-i)p T ^133 • • • P ^3U»- iP> 1 • • T^xp-Da • • P

It will be seen that (7) is identical with (6) except insofar as to all the r representing correlations between two measurements, the element / has been added, and insofar as the number of equations and of unknown quantities has been decreased by one. We may, therefore, continue a series of successive substitutions which will always result in equations of the same form. The next substitution would be

lip"" r I (p-I) p r (p-I)2p

fiacp-op — 1 — r a( p_ 1)p r ( p-,) 2p

The last substitution will give us

'"12(45. • P) — *"ia(45 • • P) **3 2(45 * • P 1

��r

��»»3 • • • p I — r a3 ( 45 ..p, r 3 2(45'-P)

Thus the coefficients of correlation between/ variables maybe reduced to those between (/ — 1) variables.

The variability p of the array of x x which is correlated to a series of values for x 2l x s . . .

p2 = Average of (*, — g x 23 . . x 2 - q x 32 . . x 3 — . . . )- = Average of (x t 9 + ?fo 3 . . x 1" + <1 i 2 32 . . x 1* + • • •

2 9\ 2 3 • • X l X i 2 ^1 3 2 • . X \ X 3 ...

I 2 ^ 1 23 • -^13 2 • • X 2 X 3 "T • • • )

By substituting the values of the types r and /a 2 for the averages of the types xx and x 2 and for q its related r, according to (5), we find

~ " 2 r l 2 3 • • r i2 2r i32 • • r i3~* • • •

4~ 2r i23 • . ; 'l32 • r 2 3~r • • • )

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