Manchester Memoirs, Vol. xlii. (1898), No. 9. 3 



where 



^- \ + 2^ .... (2), 

 the elastic constants A and /.t being those of Lame, viz. fj, is 

 the rigidity, and X is a constant such that A + |/i measures 

 the elasticity of volume. In terms of the same quantities, 

 we have 



^=^ • • ■ (3). 



If p be the volume-density of the material of the wall, 

 the equations of motion of the tube will be 



where/ denotes the excess of pressure on the inner surface. 

 Substituting from (i) in (4) we obtain 



Brdhi a_ dw\ 



dht. 



dt'^ p V dx^ "^ a dx) 



dhv p Bi(T du '^ ' 

 dfi ~ hp p\a dx 



If we now assume that tc, w, and/ all vary as 

 we have 



^\ ifr B 



u + ~w - o, 



p / 77ia p ' 



u -V 



ma p 



V I'l'ci^ P J ~ mVip j 

 Here in is a constant such that 2iT\7n is the wave-length ; 

 and c is the corresponding wave-velocity. 



It will tend to make the subsequent results more 

 intelligible if we recall the known solutions of these 

 equations when p = o. The formula (6) then gives, on 

 eliminating the ratio ulw, 



'^-V'-^^Jy'^^j) =- ' • (7), 



* The flexural terms are here omitted. They are quite unimportant 

 unless the wave-length of the disturbance be very small. 



