396 JACKSON. 



a vertical side at the extreme right, the lower and upper ends of this 

 side are denoted by u'l and w^ respectively. Suppose that is a 

 quantity in the interval 



The representation of the numbers Wie^\ . . ., Wve'^, is obtained by 

 rotating the original polygon through an angle 6. It is clear that 

 as d varies over the interval, the argument of the complex quantity 

 {wi—iVs)e^^, where s is any subscript greater than 2, has a positive mini- 

 mum. The latter statement remains true if 6 is allowed to range over 

 an interval 



V 



where e is a sufficiently small positive quantity. Now if p = re^^ is 

 in Tq, 6 does satisfy the last inequalities, at least when /• is sufficiently 

 large, and this has the consequence that as r becomes infinite, the real 

 part of {loi — Ws)e^^ remains uniformly away from zero, the real part 

 of pa{wi — Wg) becomes uniformly infinite, if a is any positive real 

 constant, and e''«(«'s— ^0 uniformly approaches zero. Furthermore, 

 the real part of Wie^'' itself remains uniformly away from zero, and 

 g_p«jio approaches zero uniformly, even when multiplied by any posi- 

 tive power of p. These facts will presently find application. 



The determinant A(p), by the vanishing of which the character- 

 istic numbers are recognized, has for the element in its 5-th row and 

 ^-th column the quantity WXyd- By substitution of (17) and (18) 

 in (16), it is found that 



Ws(ijt) = {pwtYsil], s=l,2,---,u-l, 

 W, {yt) = (pWiY" C^t- [I] + a, (piVtY [1], 



where a^ is the last coefficient aj that is different from zero in (16); 

 if every aj is zero, a^ is understood to have the value zero likewise. 

 The factor p^«, which is common to all the elements of the 5-th row, 

 5=1, 2, . . ., V — 1, does not affect the vanishing of A, and may 

 be divided out; and we will divide out also a factor p'^v^pwi^ from the 

 elements of the last row, though this factor is not explicitly apparent 

 in all the terjns. The problem is thereby reduced to that of the van- 

 ishing of the determinant 



