the Multiple Correlation Coefficient. 209 



We have 



z=.(w — uv)l y/l — u 2 Vl — v 2 , 



/. dz u — vwdv 1 / 7 (v — wu)du\ 



— + z. , = [dw — ±— t 1 — 1; 



2 w—uvl — v 1 w — uv\ 1— ir / 



so that, as before, 



2^ (i«— UU)^ 2: W — UV 



(10— wu) 2 L 



-2(,-»«)(l-^)(t,- 2(1 _J^_ 4()2j )]; 



WhenC6 ' _? (!_*&==> b. 



= ^# + 



1 



2\2 



x? !~(w — vw) 2 — (v— wu) 2 — (1 — W 2 ) 

 (w — uvy LV ' v ' v 



+ 2u (u — wu) (1 — w 2 ) — iuu(v — wu) A/(l — w 2 ) ] 



_(l-2 2 ) 2 , A(-l + ^+^ 2 -H 

 So that 



r 2 + (w-uu) 2 (l-w 2 ) 



-p (itf — iw) 1— 2 2 2 A 3 1P(1 — U 2 ) 



p,<f " " w— uu> 2c 2 (1 — 2 2 )(W — w) (w- mv) "~ 2(w— uw) ' 

 remembering that 



(l_^)(l_ t ,»)(l_ 2 2 )=A) 



R w =- Z ((l-t,>- -^!_) 

 2(w — mo)\ 7 1 — 2V 



= - ^(1--^) V w {w-uv) - (1-w 2 )* 2 ] 



2(u — vw) (u> — wu) L v y v y J 



752; 



— 7 < 7 > 



