416 JACKSON. 



exceed a quantity d e""" *^°* (2ir/;')^^5+2 ,^g j; ^ 5, by hypothesis, there 

 will be at least three of any v successive values of q for which 



' /(a:)?)„(a;)rfa- >-^e"'^cot(2,r/.) 

 I n 



It is readily seen that 



Un (x) Vn (x) dx \ < C4 e^'^cot (2V)^ 



X" 



and the rest of the divergence proof follows closely the lines of those 

 already indicated. 



It suffices novy- to give a brief description of the principal features 

 of the problem which is presented if n of the given boundary condi- 

 tions, instead of two, involve the point tt, the number fx being any 

 number less than half as large as v. The subscripts of the roots Wt are 

 to be assigned in such a way that if 6 is between and w/u the real 

 parts of the numbers iVfe^'' succeed each other in decreasing order of 

 algebraic magnitude. In the expansion of the determinant from 

 which the characteristic values are found, the largest exponentials 

 will be common to the principal terms, and so will not have an influence 

 on the vanishing of the function; the occurrence of roots will be ren- 

 dered possible by the balancing of e''"'^'^ against e''"'M+i'^, when ^u is 

 odd, and by the balancing of 6""'^'^ against e''^M+i'^ on the one hand 

 and of e"^!^-^^ against c''^^-^^'^ on the other, when fj. is even. The 

 characteristic values of sufficientl}' large index will be roots of the first 

 order, and will occur at approximately equal intervals; for odd values 

 of jj. they will be distributed asymptotically along the axis of reals, 

 and for even values of fi, along the ray arg p = ir/p. They will be of 

 amplitude or t/v exactly, from a certain point on, if the problem 

 is real at the start. The characteristic functions may be represented 

 by the expression 



(48) Un (x) = [di] e"""'/'^ -t- [d2] e''''"'*'+i% 



where di and do are conjugate imaginary ciuantities different from zero, 

 and the asymptotic representation holds uniformly throughout any 

 closed interval interior to (0, tt). It will be observed that the numbers 

 pnWf^ and pnw^+i are situated asymptotically along rays making angles 

 of p-tt/v with the positive axis of reals, whether p is even or odd. By 

 using a somewhat more explicit form for the characteristic functions, 

 it may be shown that if a series of constant multiples of them con- 



