J J 33 



if Tj, ?\„ . . . Vc, r -1-1, . . /v represents a permutation of the numbers 

 1,2, ...(7. Apparently this last minor is again a determinant B of 

 the 9*'' order of the niatrix M. Hence we have 



^,0-fl,»- . ^ 1 • • • Ap-\-\,r^ 



Kr, 



r+1 



A,,r, 



t^'~' X D 



and 



so that 



n Hu 



1 



A2f--p) 





L' ' 



^:= 



So our result is : 



W=z 



E 



2D' 



1 1 -^//dcj , 



in Ï 



and 



ökich 



bju = {~iy+^^ 



2D' 



E = 



1, 2, . . . (> 



2D"' 



wlule D represents a determinant of the {>''' order of the matrix 



ail, «12, • • • ttu 



M 



a-21, «22? • • < «27 



«5l7 «52» • • • «CÏ 



and Dj a determinant of the (<?—!)"' order of the niatri.r Mj, n^hich 

 is obtained by omitting the row aji, aj->, . . . aj, in M. 



Moreover the two determinants Dj and Dk, in the products are 

 built up from the same columns of M. 



Returning now to the coefticients ciji we have only to write 



aji 



Denoting by T), T)j, Uk the determinants in the coefficients aji , 

 corresponding to D, Dj and Dk we have 



