336 Lord Rayleigh : Problem of Random Vibrations, 



If we suppose p = 2, s = i(5 — n). Thus « = 7 makes 

 s=— 1, and infinities might occur for special values of r. 

 But if w>7, s<^, and inHnities are excluded whatever may 

 be the value of r. 



Similarly if /? = 3, infinities are excluded if n> 9, and so on. 



Our discussion has not yielded all that could be wished ; 

 the subject may be commended to those better versed in 

 pure mathematics. Probably what is required is a better 

 criterion as to the differentiation under the integral sign. 



We may now pass on to consider what becomes of 

 Kluyver's integral when n is made infinite. As already 

 remarked, Pearson has developed for it a series proceeding 

 by powers of 1/??, and it may be convenient to give a version 

 of his derivation, without, however, carrying the process 

 so far. 



The evaluation of the principal term depends upon a 

 formula due, I think, to Weber*, viz. 



u = Qjrm)*-***dw = ^*-**', . . (39) 



making f 



= ~ W 2 Jo **(?*)*'******= - if u - 



Hence u = Ce~ r2/4p2 . 



To determine C we have merely to make r = 0. Thus 



C™ 9 1 



C — u r=0 = 1 e~ p ~ x x dx — q— 2 , 

 Jo L V 



by which (39) is established. 



Unless Ix is small, the factor {J (lx)} n in (32) diminishes 

 rapidly as n increases, inasmuch as Jo(/^) is less than unity 

 for any finite Ix. Thus when n is very great, the important 

 part of the range of integration corresponds to a small Ix. 



* Gray and Matthews, loc. cit. p. 77. 



t I apprehend that there can be no difficulty here as to the differ- 

 entiation, the situation being dominated by the exponential factor. 



1 f 



