1128 



or 



Now the following relation holds good : 



n=:e 





X 



A'+h r^_^^, A^+1, r,_^,. • • • ^4,^+1, r^ 



^-,+, ' ^^'>+. 



.U,- 



in which i\,i\, . . . Vc, rc-\-i, . . .r^ represents a permutation of numbers 

 1,2,... a and the summation must be extended over all these 

 permutations. 



As 



v+i. 





.+1 





is the minor of the reciprocal determinant 



Aiu ^12, . . -Au 



A := Aou A22, ■ . • -425 



Ali, Ar2,-..A,, 



which corresponds to the algebraic complement of 



«Ir, » • • • «Ir 



''r^i , 





we have the relation 



'r+1' 



A, 



V+1 



^4.. 



1=A 



Consequently we find for JSf 



iV=A' ^~' X^ 



7-C;-l 





«bl 





«1? 



Ill 



a., 



i.e. ^ is A' '" " times the sum of the squares of all determinants 

 of the o^^^ order of the matrix 



an , ai2 1 • ' ■ oUtI, 



M=\ \ ' : 



! «pi 5 «p2 5 • • • «C 



which is formed from the coefficients of the given equations of 

 substitution. 



If we represent such a determinant of the o^^^ order in general 

 by D, we can write 



The numerator Tjk is reduced as follows: 



