330 Lord Ruyleigh : Problem of Random Vibrations, 



The lower limit for 6 is 0, and the upper limit is given by 

 cos 2 9 — a? jb 2 . Hence u = l/b, and thus 



b\ J l (b.v)J (ax)d.v=l, (b 2 >a?) \ . 



or =0, (a 2 >b 2 )) 



A second lemma required is included in Neumann's 

 theorem, and may be very simply arrived at. In fig. 1, 



Fk. 1. 



G and E being fixed points, the function at F denoted by 

 JoG/), or J \/(e 2 + f 2 -2efcos&), 



is a potential satisfying everywhere the equation V 2 + 1 = 0,. 

 and accordingly may be expanded round G in the Fourier 

 series 



A Jo( e ) + AiJi(^) cosG-f A 2 J 2 W cos2G+..., 



the coefficients A being independent of e and G. Thus 



^( 2,r JoV / (^ + /' 2 -2^/cosG)^G = A Jo(^). 



By parity of reasoning when E and F are interchanged, 

 the same integral is proportional to J (/), and may therefore 

 be equated to A 'J {e)3 (f), where A ' is now an absolute 

 constant, whose value is at once determined to be unity by 

 making e, or/, vanish. The lemma 



p J s/(e 2 +f-2ef cos G)dG = 27rJ (e)J (f), (25) 



is thus established *. 



* Similar reasoning shows that if D (g) represent a symmetrical 

 purely divergent wave, 



\ DoV(« 2 +/ 2 -2e/cosG)^G=27rJo(e)D (/) ; 



provided that/>e. 



