406 JACKSON. 



the first row be denoted by 5'i and 5'2- As in the corresponding case 

 which was discussed earlier, it is seen that neither of these numbers 

 is zero, and that each is the conjugate or the negative of the conjugate 

 of the other. We shall denote by vn{x) in the one case the value of 

 the determinant as it stands, in the other, the determinant multi- 

 plied by i. Then we have 



(34) vn{x)= Z [d't]e'"'^ti--=^), 



1=1 



where d\ and fZ'2 are conjugate imaginary quantities different from 

 zero. 



Let us examine the order of magnitude of the denominator of the 

 fraction in (32), using the expressions (22) and (34) for the character- 

 istic functions. In consequence of (20), (21), there is a real number 7, 

 independent of n, such that ^^ 



(35) R (pnWt) < (Mo+n) cot (tt/p) +7 



for t = 1 and t = 2, and a fortiori for ^ = 3,. . ., v; the constant 

 fM)Cot{Tr/v) might be merged with 7. Hence there exist numbers D, 

 D', and c, all independent of n, such that 



I Un{x) I < De'^ ^cot (x/.)^ I ^^(^^) [ < 2)'e" ('^- ^) ^^o* ('^Z"), 

 and 



^^^^ \ /q '^n{x)Vn{x)dx < cfi"'^ cot (,rA) _ 



Now let us define a function /(.r) by the formula (14), denoting by q 

 for the present an arbitrary positive integer, and seek information 

 about the corresponding value of the integral in the numerator of (32), 

 with the aid of the expression (34) for vn{x). By integrating by parts, 

 as at the corresponding point in the preceding section, with attention 

 to the special values of /(.t) and its derivatives at and tt, and supple- 

 menting this process by the application of (20), (21), and (35), it is 

 seen that 



/ V(a;) e"""" ^"""^^ dx — [ -^— ]' e"" 



Jo •'^ ^ \pnwj 



_ f 1 Y+2 



^ J^ pTlv cot {it/ v) 



where Ci, like every quantity denoted by the letter c with a subscript 

 throughout the remainder of this section, is independent of n. 



32 By R{z) is meant the real part of z. 



