the Multiple Correlation Coefficient. 207 



cosC = 3 r 12 . Hence the r 12 , r 23 , r 31 will be found from the 

 forinulse for the polar triangle, viz. : 



cosB cos C+ cos A 



or 



cosa = 



23' 



sin B sin 



1^23 + 2^31-3^12 



(2) 



(ii.) From the relation 



l- 1 K^=(l-^)(l-3^) 



= sin 2 b sin 2 C, 



it is clear that {R 2 z is the cosine of the perpendicular from 

 A on BO. 



(iii.) Let us write 



so that 

 x = 



r 23 = cos a = a, x r 23 = cos A = #, 

 r n = cosb = v, 2 r 3i= cosB = z/, 

 r 12 — cos c = w, 3 ?' 12 = cos 0=z, 



and u = 



3 i=Y, > 

 Ri 2 = Z, J 



i-R 23 — 

 2 R 31 = Y 



(3) 



u — vw 



XArVZ 



y/l-v 2 ^1-w 2 

 in addition, let 



A = 



1 



12 



1 ^23 



r 3l r 



\/l-f \/l-z 2 ' 



= 1 — u 2 — v 2 — W 2 + 2uvru, 



(4) 



and 



A' = 



1 

 ■3^*12 



■3^12 — 2 r 13 

 1 - 1^23 



_ l r 23 1 



= 1 — x 2 — y 2 — z 2 — 2xyz, 



(5) 



So that sin a/sin A = V A/ A', 



A=(l-u 2 )(l-w 2 )(l-;u 2 ), A' = (l-y 2 )(l—z 2 )(l-u 2 ) 



(iv.) It is well known that in normal distributions the 



^2 

 standard deviation of r 12 is given by ^ 



(1 ^ and the 



correlation between deviations in ?* 23 and r 13 is 

 R uv = w-uv&/2{l-u 2 )(l-v 2 ). 



