the Multiple Correlation Coefficient. 211 



so that dA = udu_ juh zdz 



2A~ l-w 2 + l-v 2 + 1-z 2 ' 



Squaring, summing for all samples and dividing by their 

 number, we find on using (7) and known results that 



nSi 



= u'+V + ^ + 2t£t;( 2(1 ^ M ^ t ^ )--iiV-f;V > 



(11) 



4A 2 

 and the right-hand member reduces to 



u 2 + v 2 4 w 2 

 when we substitute 



z = (io — uv)\\/l — U 2 \/\ — V 2 . 



S ° that CT A= X /U 2 +V 2 + W 2 



2A v/ n 



But (l-w 2 )(l--X*)=A. 



H ence X<£X M <fa _dA 



1-X 2 + 1-M 2 ~ 2A' 

 and therefore ,,2 . ,.,2 _ X2 



gives the correlation between errors in r 2S and R lt23 

 (ix.) Similarly from the relation 



A' = (l-^ 2 )(l-t/ 2 )(I-^ 2 ), 

 and equations (6) and (7), we deduce 



(13) 



%*_ \/x 2 + f + z 2 



2A' s/n ' ' " " 



and then using (1 — ,^ 2 )(1 — X 2 ) = A', 



we find ? ,2 1 J_v2 



(x.) To find the correlation in errors of two multiple 

 correlation coefficients X = E 1 . 23 and Y = R 2 . 3 i > we have 



(1-Y 2 ) = (1-™ 2 )(1-* 2 ) 



or Y^Y wdw xdx 



l_Y 2 ~"l-w 2 + l-# 2 ' 



