332 Lord Rayleigh : Problem of Random Vibrations, 



Taking first the integration with respect to U we have 

 by (25) 



1 i 2lt 



- ) ddiJoUsx) = J (^) Jo(W> 



and thus p^. fi> y = i 8 4 ,j i(ra)Jo(Wi[fl(Wi < (27) 



Jo 

 The method can be extended to any number (n) of stretches. 

 Beginning with the integration with respect to 0„_i in (26), 

 we have as before 



— \ (Wn-i = 5- 1 ^„_i 1 da J ! (ra) J (s»#) 



*• * ■ %J o 



Jo 

 The next integration gives 



r^—y 1 I dO n . 2 dO n -i = r I Ji(r#)J (4#) Jo(^-i^jJo(^-2^')^ 

 and so on. Finally 



P„(r ; /„ ? 2 , ... /„)= —pnftV.. «»!«»,. ..iW,^, 



(■»oo 

 Ji(^)J (Z 1 d')J (^^) ••• Jo(Z»#)d#, . (28) 

 . o 



— the expression for P„ discovered by Kluyver. 



It will be observed that (28) is symmetrical with respect 

 to the Ts ; the order in which they are taken is immaterial. 



When all the Vs are equal, 



F n {r; l)=r\ J 1 (^){J (^)}^- • • (29) 



Jo 



If in (29) we suppose r — l, 



F n (l; V)=-C {J (lx)}*dJ (k:) 



'0 



{J,(k)}- +1 • 

 n+1 



n + 1 



(30) 



so that after n equal components have been combined the 

 chance that the resultant shall be less than one of the com- 

 ponents is l/(?i + l), an interesting result due to Kluyver. 

 The same author notices some of the discontinuities which 

 present themselves, but it will be more convenient to consider 

 this in a modified form of the problem. 



