392 JACKSON. 



to remain finite as n becomes infinite. Let xo be any particular value 

 in the interval < .tq < t. Modifying the form of (7), we may write 



Un{xo) = e- p""^ -{- 2 c''"""^ sin (-^ pnXo - '^j. 



The sine in the right-hand member may be zero or nearly zero for 

 some values of n, but there are certainly infinitely many values of n 

 for which this is not the case. For 



1^^- P„ + 1 ^0 - gj - ^^— Pn .To - g J = ^0 + — (en + 1 - en) .To, 



which ultimately becomes practically equal to .tq, so that the two 

 parentheses on the left-hand side can not both be nearly equal to 

 integral multiples ^"^ of tt. It follows that for infinitely many values 

 of n 



V3 



'sm, . ^,...>, „ 







Un{xo) I > €36^""''°, 



and 



I anUn{,Xo) I > 



Q+2 



Pn 



where Co, c^, and C4 are all positive and independent of 71. As the right- 

 hand member of the last inequality becomes infinite with n, the proof 

 of Theorem II is complete. 



It may be remarked that no use has been made in this demonstra- 

 tion of the fact that all the characteristic numbers are given by the 

 asymptotic formula that we have employed. The proof would be 

 unimpaired if there were infinitely many others. It is sufficient to 

 know that there exist an infinite number of real characteristic values 

 distributed in accordance with the formula. And this latter fact is 



17 More precisely, if 5 is the smaller of the quantities .xo, ir — xo, the index 

 n may be made so large that 



and then the difference between the two parentheses is between ^5 and tt — ^5, 

 and one or the other of the parentheses must itself differ from the nearest 

 integral multiple of tt by at least \h. 



r5 



0:0 <-, 



