the Multiple Correlation Coefficient, 215 



Now, by (17) a, /3, y are of order while a u , <r n <r„ are 



1 1 n 



o£ order — - — . Hence correct to - we shall have 

 \/n n 



OU ^dv ' ow 



+ ^r^l^ + ^l^+cr^ l^ + 20-.cr^^X 



2 L o^ o» ow o&ou 



+ 2w -^ +2wV '"^J • • • • ( 22 ) 



Since we are in this section neglecting l/?i 2 , we take 



a= —u(l—u 2 )J2n, 



o; u ={l—u 2 )/\/n, 



r u ; = w-uvA/2(l-u*Xl-v 2 ), 



where A=(l-u 2 )(l-X 2 ). 



X==/(w, v, w) = vv 2 + w 2 — 2uvw/\/l — it 2 , 



we have easily 



X(l — u 2 ) -~ — = uX 2 — vw, 

 qu 



~K(l—u 2 )^— =v — wu, 

 OV 



X(l — u 2 )^ — =w — uv. 



QW 



X 3 (l -u 2 ) 2 1^ = X 4 (l + 2u 2 ) - 2uvwX 2 - v 2 w 2 , 



x*(i-u 2 )ig^ 2 , 



X 3 (l-u 2 )|^=*, 2 . 

 X\l-u 2 y ^^=vw(v-wu)-(w-uv)X 2 , 

 X 3 (l-u 2 ) 2 ^^- = vw(v-wu)-(v-wu)X 2 , 



X 5 (l-K 2 ) ^-^- = -^. 



* We prove formally in a later section that the terms arising from 

 the cubes and higher powers of the deviations make contributions of 

 order 1/n 2 . 



