112 



CHAPTER IE. 



where 



92) 



in which 



_ - 71 



i 1, . . ., m 

 Z = l, ...,,7 







tt 7 : 6 * 



.72*2 



~b q denote the integers 1, . . ., I 1, I -|- 1, . . ., q. 



Since the determinant a*^ | =4= 0, being the determinant of S, we have 



Hence the determinant 92) vanishes for '!,..., m and for 

 & 2 , & 3 , . . ., & 9 an arbitrary combination of q 1 distinct integers <| q. 

 If # = 3, we have reached the relations 95) below. If q > 3, 

 we denote by C\ b * k the adjoint of a! 6 * in the determinant 92) and 

 consider the following expansions: 



93) 



C ib * = 0. 



8=2 



Of these consider the mq equations in which i, j s , . . ., j q have fixed 

 values chosen arbitrarily from 1, 2, . . ., m, but such that j 3 , j 4 , . . ., ^ 

 are not all equal, and & 3 , . . ., fc g fixed values chosen arbitrarily from 

 1, 2, . . ., g, while lastly j 2 takes the values 1, 2, . . ., m and & 2 the 

 values 1, 2, . . ., q. Since the matrix 



-2, - .., 2 



comprises g 1 rows of the matrix of /S, not all of its determinants 

 of order q 1 are zero. Hence the q 1 determinants 0, which are 

 the same in each of the mq equations 93), must be zero, viz., 



94) 



where c 3 , . . ., c q denote any q 2 distinct integers < q. 



If q = 4, we have reached the relations 95) below. If q > 4, 

 we proceed as before. After q 2 such steps, we reach the set of 

 relations 



a Tk % 



95) =0 



r, s, 



