the Multiple Correlation Coefficient. 219 



We have, as before, 



(X + dX) 2 = cj>(u + a + du, v + @ + dv, w + y + dw) 

 or ^ j 



X 2 + 2XdX + tfX 2 = X 2 + (a + ^)^ + ..- 



i[(« + ^n + ...]. 



On summing for all samples and dividing by the number 

 of samples, we have 



A QU Ov Qw 



Hence correct to 1/n, writing X 2 for tj>, 



Y 2 v2 ^2 M (l ~~ ^ 2 ) ^ (vX 2 — via) v( I — v 2 ) 2(v — uw) 



A+2, x - 2n~~ 1-u 2 2^ (1-u 2 / 



2«?(1 — IV 2 ) (W — 11 v) 1 rnvo -.ooo -i 



o n 2A + o- [2X 2 (1 + 3u 2 ; - 8uiw] 



2?i(l— ?r) 2n L v 



«(1 - u 2 ) n(l-u*) 



(1 — -^ 2 )(1t 2 ) / _ in; 1-X 2 \ 4?q?-2w — 2t**w 



~nT " \ * "*" l 3 ? / _ (I-?/ 2 ; 2 

 (l-r 2 )(l-2i- 2 ) / _ tw(l-7/ g )(l-X g ) \ 2?v 

 n V 2{l-v 2 ){l-w 2 ) ) T^ 2 



(l—u^Xl — w 9 ) ( _wu 1-X 2 \ (-kwu — 2v-2u*o) 

 n V 2 1=^2/ (1-u 2 ) 2 ' 



(l-u 2 y 

 reducing to X 2 — X 2 + 2x 



i X 2 {(1 — u 2 ){l 4-2u 2 ) + 2u 2 (v 2 + w 2 ) -3uvw~u 3 vw\ 

 „ 1 ^ <? — 5 v 2 + 2v 4 - 5z^ 2 + 2w± + 2 - 2w 2 + 4u V 



f - 2 w 2 (> 2 + w 2 ) + u vio( 1 + 4w 2 — 5 v 2 - 5w 2 -f 2 www) 



X2(1_ w 2 ) (1 + 2 w 2 )+ 2^ 2 X 4 (1-h 2 )-3ww;X 2 (1-w 2 )) 



+ 2(l- M 2 ) 2 X 4 + 3w^(l-u 2 )X 2 -5(l- w 2 )X 2 l 



2u 2 (l-u 2 )X 2 + 2(l-^ 2 ) ) 



= na i ^(l-u 2 )(2X 4 + X 2 (l-f2^ 2 -3^^ + 3^^-5-2z/ 2 )+2)f 



= -(l-X 2 ) 2 . 



