the Multiple Correlation Coefficient. 217 



Secondly, 



tJI= —uvw(1 — u 2 ) — v(v — wu)(l — v 2 ) — iv (iv — uv)( 1 — IV 2 ) 



+ uv 2 w(v — ivu) + 2v(uiv — v) ( 1 - iv 2 ) — 2w(iv — uv) (1 — w a ) 

 -ruvw 2 (w— uv) — v 2 iv 2 (l~u 2 ) + wu(y — ivn) + uv(w — uv) 

 = uviv(l +ii 2 — 2v 2 — 2w 2 — uvw) 



-3v 2 + v 4 -3w 2 + w* + 3v 2 w 2 -u 2 v 2 -u 2 w 2 ; 

 or after some reduction, 



UG?=(1— M*) 2 X 4 + (1-h*)X 3 (2i«w-u 2 -3) + (l-?* 2 )(utw + v'u-*). 

 Finally, 

 $£= v 2 + w 2 - v 4 w 2 - v 2 w± + i«m( - 2r« 2 -w 2 + 2u V 2 + 2uui0) 

 = (1 - u 2 ) X 2 - tV(l - i* 2 )X 2 - uvw(l - u 2 )X 2 

 = (1 - y*)X"(l - v 2 w 2 - uvw) . 

 So that 

 fX 3 (l-u 2 )2?i 



-^X 4 -h^X 2 + ^ 

 = {l-u 2 )X%u 2 X 2 -2uvw-t-l+u 2 ) 



+ (l-u 2 )X 2 [(l-u 2 )X± + (2uvw-u 2 -Z)X 2 + vw(u f «w)] 

 + (1 — ii 2 )X 2 (l - v 2 w 2 — uvw). 

 .-. 2n|X=l-'-X 4 4-X 2 {-2^^ + l + ^ 2 + 2i^^-3-^ 2 }, 



= 1 + X 4 - 2X 2 , 

 or 



£=X-Y = ii 



2nX 



i. e. the mean value o£ R r 23 in many samples is always 

 greater than the value in the sampled population, the 

 principal term in the excess being the positive quantity 



{ ■*- -"1-2.3; 



2nR v2 , 



The form of this expression is highly significant. It is 

 not only for small samples that the mean value differs 

 sensibly from the value in the general population. If the 

 value of R in the sampled population is small, the mean 

 value deduced from many samples may considerably exceed 

 the true value even in samples of large size. For example, 

 consider samples of 1000 taken out of a population for 

 which R=-01. 



Phil Mag. S. «. Vol. 34. No. 201. Sept. 1917. Q 



f=X-X^(l— §!£, .... (23) 



