342 Lord Rayleigh : Problem of Random Vibrations, 

 Thus 



%-&{*-*\r-l\ + \r-U\y . . (61) 



■expresses the complete solution. When 

 r<l, dFz/dr = r 2 /2l% 

 •M>r>L d? 3 /dr= {Mr-r 2 )/4l\ 

 /•>;)/, dF,/dr = Q. 



It will be observed that dP 3 jdr is itself continuous ; but 

 the next derivative changes suddenly at r = l and r = 3/ from 

 one finite value to another. 



Next take n = 4. From (59) 



d? 4 2rC«dx. . 



—j— = — 1 ., sin rx sin 4 Ix, 

 dr ttI*J x* 



and 



— i"9 1 - " / ) = —f\ \ ~ sm ra sin 4 /a? 

 ar* \r dr J it I J a; 



— ~r - m j ~{sin (r + 4/)./' + sin (V — 4/).r 



— 4 sin(r + 2/) a* — 4 sin(r— 2l)%-\- 6 sinrA*} 

 = i 1 f{ 1 ± 1_4+4 + 6} = i 1 i {3±1+4}, 



the alternatives depending upon the signs of r— 4/ and r — 21. 

 When r < 2/, - 16/^f- ^ = 6, 



4/>r>2/, _16^^ 2 2 f 1 ^) = -2, 

 ar'Vr dr / 



and when r>4/, the value is zero. In no case can the value 

 be infinite, from which we may infer that 



dr \ r dr J r dr 



must be continuous throughout. 



From these data w r e can determine the form of dPjdr, 

 working backwards from the large value of r, where all 

 derivatives vanish. 



(4Z>,>2 -16^g5) = _2(,-4*), 



(2l>r) -16^g^)=6(»-20 + 4fe6r-8/, 



