VAN DER VEIES. — MULTIPLE POINTS OF TWISTED CURVES. 479 



row and the kth "column, then k — 1 constituents diagonally, beginning 

 with the vth row and the first column, and finally /a — k constituents 

 from the last /a — k rows and columns. These last /x — k constituents 

 must be constants from the main diagonal, otherwise a zero-coefficient 

 will be brought in, causing the product to vanish. This choice will give 

 us the terms or S of highest degree in s. For, if we begin with the con- 

 stituent of the first row and (k + l)st column, we can bring in one more 

 constituent in our second group, and one less constituent from the main 

 diagonal in our last group. Doing this we diminish the degree of s by 

 unity in each of the first v — 1 rows, and bring in one new term in s v ~ *' ; 

 this does not, however, compensate for the loss of s v ~ l . A similar loss 

 will occur every time we move one place to the right. If we begin with 

 the constituent of the first row and (Jc — l)st column, we do not change 

 the power of s in the first v — 1 rows, but lose a multiple of s v ^~ k in the 

 second group, and only bring in an additional constant constituent in the 

 third group. A similar loss will occur whenever we move a column to 

 the left. Nor can we obtain a term of higher degree in s by taking any 

 other combination of constituents of our scheme. For, if one term of a 

 determinant is known, another term can be obtained by interchanging 

 any two rows or columns of the determinant and taking the constituents 

 corresponding to those used in the determinant before it was so changed, 

 e. g. considering the two rows and columns ; — 



if a and d are factors of the known term, we can get another term by 

 substituting b and c for a and d. This process can be repeated until any 

 desired term is obtained. Compare the term we have chosen with any 

 other term that can be obtained in this way. If a and d are both in the 



