Manchester Memoirs, Vol. xlii. ( 1 898), No. 0. 5 



Let us now suppose that the tube contains an elastic 

 fluid of density o^. The pressure/ is subject to an equation 

 of the form 



i) jd'~p d^'p idp\ . - 



where r denotes the distance from the axis, and c^ is the 

 velocity of waves of expansion in an unlimited mass of 

 the fluid ; viz., if we denote by k the cubical elasticity of 



the fluid, we have 



Po 



■ (16). 



u 





p oc ^""rcH.;^ 





(15) reduces to the form 





^V ^dp „ 

 dr^ rdr 



(17), 



where 





,.=..(. -J) . . . 



(18). 



The solution of (17) which is finite for r=o is 





p^CIXvr) 



• (19), 



where the function /^ is defined by 





2^ 2^.42 



■ (2°)- 



Now, from the hydrodynamical equation 





'V/2 L^J_ • • 



• (21), 



we have 





mVp,;ia^yCi:{va) 



(22), 



whence, for the value of/ to be substituted in (6), 



p = 7nH-p^a jTi — ■..w . . (22). 



The complete solution of our problem involves the 

 determination of three classes of vibrations ; viz. (I) the 

 sound-waves in the fluid as modified by the yielding of the 



