412 JACKSON. 



where the first term on the right is a particuhir value of the logarithm, 

 and n is any integer. As in the case previously treated, the logarithm 

 is a pure imaginary. Consider, for example, the relation between 

 the determinants 5i2 and Si^. The latter may be regarded as obtained 

 from the former by replacing w'3 in the first column by w^. But it 

 may also be regarded as obtained by replacing each of the roots from 

 W3 to %v by the root next to it cyclically in the negative direction 

 (whereby W3 goes into one of the roots with a subscript greater than 4, 

 and Wi goes into w-z) and then interchanging columns. This process 

 amounts to a multiplication of each row of 812 by a root of unity, and 

 so to a multiplication of the whole determinant by a quantity of the 

 same sort. Similar reasoning applies to f 12 and f 13, and it follows that 

 the ratio 613^13/512^12 has its absolute value equal to unity, and the real 

 part of its logarithm equal to zero. The successive values of p in 

 (42) are not real, because of the factor e'"^", but follow each other at 

 equal intervals along a ray inclined to the axis of reals at an angle of 

 tt/v. The correspondence between these values and the roots of the 

 equation Ai = is established as in the earlier cases. 



The roots of Ai = in the region T2V-1 are similarly determined. 

 There will be infinitely many of them, situated asymptotically along 

 the ray arg p = — tt/v. But since the original problem involves 

 p only in the j'th power, as has already been pointed out, the existence 

 of roots p along this ray implies the existence of roots pe-'^*^" which 

 will be in the region To, at least from a certain point on, and so will 

 be among those already found. Thus the consideration of the region 

 T^v-i adds only a finite number to the list of independent character- 

 istic values. Hence: 



The characteristic values of the system (15), (39), are simple roots 

 of the determinant equation from a certain point on, and are expres- 

 sible in the form ^^ 



n-f- en — 1110 



Pn = ■ /.^ , r ^ ' n = g,g+l,g-\- 2,---, 



sm {zir/v) 



where yuo is real and independent of n, g is a suitable integer, positive, 

 negative, or zero, and 



lim ^ 



en = U. 

 n= 00 



35 It may be that there are infinitely many characteristic values just inside 

 the lower border of jS-jv-i, and then the formula will give instead of these the 

 equivalent values in Si] but the latter will serve as well and will be more con- 



