1 2 Lamb, Velocity of Sound in a Tube. 



restriction is abandoned we must fall back on the general 

 equations of Elasticity.* 



In a usual notation, the equations of motions of an 

 elastic solid free from extraneous force are 



where 





dx dy dz ' ' ' ^^ '' 



If the time factor be ^*""'*, it is known that the general 

 solution of these is of the type 



I ^^ \ dl \ dl 



''^-J'Jx'-"' ^=-1^^ + ^' ^=-/P^ + "'- (43), 



where S is the solution of 



(v2 + ^2)a = . . . (44), 



and u, V, w constitute the general solution of the system 



diL dv dw 

 dx dy dz~ 



(45-)t 



The constants h, k, which appear in these equations, 

 are defined by the relations 



ii' = mh^p!{\+2fx), k^ = mh''plfx . . (46). 



In the case of symmetry about the axis of x, the 

 equation (44) takes the form 



d^ d'-h I dh ^^^ 



^2 + ^2 + ;^+/^^^ = o . . . (47), 



where 



r={y^ + zy. 

 We may also write 



- = ^-^^^x. --j^y -=^:i, . (48), 



* It has not been found possible, in the investigation which follows, to 

 preserve altogether the notation of the previous part of this paper. 

 t Cf. Proc. Loud. Math. Soc, t. xiii., p. 192 (1882). 



