6 Mr Berry, On the evaluation [Oct, 31. 



Then by the ordinary rule for Jacobians we have 



9(<j> 12 , ... ^n-i,n-iy •■• F\n, ••• -^w-i, n) 

 v \Ti2i '" fn—i, n) 



^V 12) •• • f n—i, n) 



y 12> '•• r n—i,n) /q\ 



3<V r' ^ ^ '" 



^ V 12 j ••• ' n— \,n) 



By direct differentiation and substitution, followed by some 

 obvious reductions, we find that the Jacobian on the left and the 

 first Jacobian on the right are respectively equal to 



(- l) n ~ l J lt and (- 1)4 («-D (•*-* J 2 , 



where ^ J(4^,-^, n -i) (4)> 



o (r I2 , 7* 13 , . . . T /l _ 2) n— i) 



and 



(5). 



J- ^ ymy r 2n> • • • ^n— 1, w) 



V in > ** 2n » • • • *" w— ij n / 



It follows that the required Jacobian is given by 

 V^r"""^ 1 '"? = ( - 1) " n ~ 1) (n - 2)+w - 1+i(w - 1) -W = W- • -(6). 



v V 12' '13> ••• '71— 1> 71/ 



§ 4. Evaluation of J x . 



Jj is a Jacobian of order %(n — l)(n— 2) but it is not a 

 determinant of the type J for the case of n — 1, since the quantities 

 r' which occur in it are formed from the determinant of order n, 

 not from that of order n— 1, and they are consequently functions 

 of the quantities r in (i = 1, 2, ... n- 1) as well as of those quan- 

 tities r, which occur as independent variables in the differential 

 coefficients. 



To prevent confusion and to avoid undue multiplication of 

 suffixes let us use the German letter dt to denote the determinant 

 of order n—1, 



-!■> r vi> ••■ ?'i, n—i 



corresponding to the determinant R of order n; and the cor- 

 responding German letters %, %j, vy for the quantities correspond- 

 ing in this case to R { , R^, r tj as already defined for the case of the 

 determinant of order n. It will also be necessary in the course of 



