394 
Oil the Partial Correlation Ratio 
R 
where b^t = ^'and Rpq is the minor with its proper sign of the element in the ^^th 
row and qih column of the determinant 
R = 
.(19), 
1 1\~2 ■ ■ • ''iin 
V21 1 ... r^fi 
^ mi ''m2 • • • 1 
while the maximum correlation between any linear function of 
^1 is 2,3, ... m^i where 

~ 2,3, mRi^ — t5 — ^12^2 + ^is^'is • ■ • "V &jm ''iOT (2^)' 
S { J?i2 {Xi - Xi) (X^ - X.,)} S {/ii3 {x\ - Xi) (*'s - Xx)} 
_ h 1? -J- /) 
Jy CTj O"^ iV CTi CTg 
aSi Xi) (if'TO '^-'m)} 
+ . . . + (21). 
Subtract (21) from (17), noting that n^2= S {iho, m], etc., we obtain^ 
3...m 
(2, 3, ,.. mHi' — 2, 3, . mMi-) Nai' 
= S {Ho, , {Xi - 2 .,„t*'i)-| + »S' \ H,o — 6,2 (Xi - X-,) {X^ - X^) 
2...m 12 (, 0"2 
+ . . . + iSi j Ji]„, by„i (*'x — aij) (a;„i x^n) 
— *S „i (cCj — 2..:nv^l)' } + i ?<2 ^12 (^2 ~ •^2) (2.. m*l ~ •''^1) 1' ~ • ■ • 
2...OT 2( 0-2 
+ »S-|7;„j bj,n(^x^ij ■*))!) (2... ^1) 
= S 
2...);( 
= s 
2...m 
m y ^ III 
(■^1 2...»i'^'i)"' "I" '"2...1H ^12 ('^'2 -^a) (2...))i"''i -^i) 
in, (2 Hl"^! "^l) )2 iiv^i ^\ ^12 ("^2 "^2) 
cr., 
•(22), 
using (IS) this equation becomes 
(2,3, ..mHi 2,3, ...111-^1') ^ ^i" ~ S {>l2,..ni (,2...mf^l ^1) (2 , m^'i -^1)} 
2. ..VI 
= S ["2,..m (2...m«l - -Yi)2} + S {%..,„ (Xi-a7i)(2...m«i-^l)} 
2...m 2...m 
(23). 
* By an extension of the notation described at the beginning of this section S3,. „j denotes a 
summation with regard to the variables .1-3, .rj, ... x^^; Ki,2,...,„ is the frequency of a particular com- 
bination of the characters Xi, x-^, ... .t,,j while ny^ is the frequency of the combination xi, x-^. 
