208 Dr. L. Isserlis on the Variation of 



Making the third variable constant we have immediately 



2 2 or 2 2 = <^-\ 



We must now show that 



the formula suggested by the polar triangle. 



For 



sin a sin A 



sin b sin B ' 



or 



Sothat udu vdv xdx 



1-u 2 l-v 2 ~~l—x 2 1-f 



Squaring and taking the mean for many samples, 



u 2 + v 2 - 2uvR uv = x 2 +y 2 - 2^yK^. 

 But 



T) UV & /I 2\ 



B "° = "'- 2(.l-^(l-^) ="-Kl-^) 



_^ z + xy (x + yz) {y + zw)(l—z 2 ) 



\fl-x 2 */l-y 2 2 ^l-y 2 s/l-z 2 \/l-z 2 ^1-x 2 

 = [2{z + xy)-(a:+yz)(y±zx)]l2 \'l-x 2 s/l-y 2 . 



Hence 



2^R jy =^ 2 -j-3/ 2 -^ 2 -^+ — ^ ===[2(s + fly) 



vl-«r v 1 — y 



-(x+yz)(y + zx)] 



- r 2,, y2 o+y^) 2 (.y + *g) 8 



-*+# (W)iW) (i-^ 2 )(i^ 2 ) 



whence ^ 



J xy 



2(l-^ 2 )(l-2/ 2 ) 



(v.) To find the correlation between the correlation 

 coefficient r 13 and the partial correlation coefficient 3 r 12 , 

 i. £. to find ll V g. 



