Mr Berry, On the evaluation 



[Oct. 31 



By exactly similar work we find that 



i=n-l 



Rj = (i - n-n)- 1 • n (l - r\ n ) . [gy ,_, 



and 



i=\ 



(8), 



R jk = (1 - r%)~i (1 - rVT 1 n (1 - r\ n ) . [% k ] r={> (9). 



Whence 

 Hence 



r 'jk=[Xjk]r- P (10). 



t y 12> • • • r n— 2 , n—i) v \Pi2> • • • Pn—i, n— 1) 



v \Pi2> • ' • Pn—2, n—i) v v 12 > • • • r n— 2, n—i) 



V \pl2 y ••• Pn—2, n—i) 



...(11). 



Since 



d( Pl 2,...p n -2,n- 1 ) = ^-VJJ-I 1 



d (r 12 , . . . r w _ 2j n _ 2 ) i>j=1 ' V(l - r 2 in ) (1 - r%) 



n (i-»- fe ) 



i=l 



-(» 



But the Jacobian on the right-hand side of equation (11) is the 

 Jacobian for the case n — 1 of the same type as J, the quantities 

 r being replaced by the corresponding quantities p ; so that by 

 hypothesis (equation 2 of § 3) its value is 



n-l 



(_ l)*(n-i)(n-2) [gt^^-3)/(n^)^] r=p . 



1 



Using equations (7), (8), (11) and reducing we obtain 



(_ 1)J(«-D (n-2) J 1 = U(1- r\ n ) . i#» <«-»>. 



n-l 



n Ri 



i 



—\n 



,.(12). 



§ 5. Evaluation of J 2 . 



It is convenient first to evaluate instead of J 2 the correspond- 

 ing quantity J 2 ', obtained by interchanging dashed and undashed 

 letters. 



We have first to evaluate 



dr'- 



-j^f (t,j = 1,2, ...,»- 1). 



jn 



Now 1 - r'\ n = (RiR n - R\ n )jRiR n = R . iR n /RiR n 



where iR n denotes the second minor in R, obtained by erasing the 

 ith. and nth. rows and columns. 



