212 Dr. L. Isserlis on the Variation of 



Multiply by dX. and sum for all samples as usual. 

 Thus 



Y(l-X )R XY _^ ^ + * 2X^— 



which reduces to w 2 + z 2 



Rxy= 2XY ^^ 



(xi.) Finally to determine the correlation between errors 

 in v = r 31 and y = 2 r 3 i. 



We have l_X 2 = (l-u 2 )(l-£ 2 ), 



i_z 2 =(i- w 2 )(i-2/ 2 ); 



so that 



XdX zdz vdv 

 1-X 2 l-* 2 1-v 2 



and 



ZdZ udu ydy 

 1-Z 2 1— u 2 1-y 2 ' 



Multiply these results and sum for all samples. Using 

 the results previously obtained, we find 



_ Y / ^ 2 + ^ 2 -X 2 \ Z^(^ 2 4-r-Z 2 ) . 

 WA \ 2uX ) 2Zz 



or 2^IC, = X 2 + Z 2 -^ 2 -zt; 2 - W V, 



which can be put in the symmetrical form 



2^R„ y =X 2 + Y 2 +Z 2 - w 2 -i^ 2 - A - 2 -^ 2 . . (16) 



(xii.) We collect the above results into a single table in a 

 form suitable for reference. 



Let A = l--r* 2 -r> 3 -^i + 2r 12 r 31 r 23 



and 

 Then 



A' = I - !>!, - a rj|; - 3 »ti-2 d*-^) ( 2 r 31 )( 3 r 12 ) . 



R r r 



' 23) ; 13 



= , 'i 2 - > vi3 A / 2 ( 1 - 4.x *- - '-y 



I r 23» 2^8] 



=r» r «-(iV^ , >») A '/2(i-i'4)( l -i'W/ • («) 



'23> 1/23 



= ( 1 R^ + 2 B| I + 3 K? 2 -r? 3 -,f 2 - 2l | 3 - 3 r? 2 )/2r 23 . ir23 . 



. , . (16) 



