we get 



334 Lord Rayleigh : Problem of Random Vibrations, 



When we substitute for J (Lv) from (34) and for Ji(rx) 

 from 



j x dx cos I — rcc \ cos \t~~^ x ) > 



which becomes infinite with #, even for general values of 

 r and Z. 



So much by way of explanation ; but of course we do not 

 really need to discuss the cases n=l, n = 2, or even ?i = 3, 

 for which exact solutions can be expressed in terms of 

 functions which may be regarded as known. 



For higher values of n it would be of interest to know 

 how many differentiations with respect to r may be made 

 under the sign of integration. It may be remarked that 

 since all J's and their derivatives to any order are less than 

 unity, the integral can become infinite only in virtue of 

 that part of the range where a: is very great, and that there 

 we may introduce the asymptotic values. 



We have thus to consider 



^^W = y o "^^ +1 J/(™){.To(te)}». • (35) 

 For the leading term when c is very great, we have 



=\/© cos G ,r - ? -^ 7r )' • • (36) 



iU')r= (§})*«*§*-*), (37) 



so that with omission of constant factors our integral becomes 



( dxx p+ >~' n cos (^TT-rx- *p7r)cos*/|7r- lx\. (38) 



In this cos n (;j7r — Ix) can be expanded in a series of cosines 

 of multiples of (\ir — fa?), commencing with cosn[^7r— Ix) 

 and ending when n is odd with cos (\tt — Ix), and when n is 

 even with a constant term. The various products of cosjnes 

 are then to be replaced by cosines of sums and differences. 

 The most unfavourable case occurs when this operation 



