210 Dr. L. Isserlis on the Variation of 



(vi.) To find the probable error of the multiple correlation 

 coefficient R123, we use the relation 



l-X 2 = (l-?; 2 )(l-z 2 ). 



Differentiating, 



XdX vdv zdz 



whence, using (7), 



X 2 ^ 2 __v* z 2 2vz{_ vz\ 

 (1-X 2 ) 2 ^x- n + n + n \ 2/ 



or nX 2 Sx 2 2jL 2 2 2 



(i-x 2 ) 2 =t; + z vz 



= l-(l-v 2 )(l-z 2 ) 

 = 1-(1-X 2 ) 

 = X 2 , 



so 



that 



tn^y 



1 R2 

 2 = iZfk? (8) 



(vii.) Correlation between deviations in R123 and r 13 or 3 r 12 

 X^X vdv zdz 



l^T 2 T^~T^z~ 2 ' 



squaring and taking mean 



X 2 + v 2 -2vXR vX = z 2 , 



or „ X 2 + ^ 2 -* 2 



Kxu== 2t,X ' 



Similarly ^*±£z* 



RB^r, =(^,3+^3-3^)/2^^,3- ■ ■ W 

 1*23, lo 



R B 1 ,3,^ 1 3 = ( R ^ + 3^-'i3)/2 3 '- 12 Rl- 2 3- • • ( 10 ) 



(viii.) We can now find the probable error of A from the 

 relation 



A = (l-w 2 )(l~t> 2 )(l-2 2 ), 



