29-4 Prof. Challis on the Hydrodynamical Theory of Vibrations. 



82 

 Now let k— = the constant a 1 . Then by integrating, 



mpV = a!mp-\-C ; 



and if C be determined so that V=0 where p = l, we have 



pY= a '(p-l); (14) 



from which equation, it may be noticed, the variable m has dis- 

 appeared. This result gives the relation between the velocity 

 and density, on the supposition that the density is propagated 

 uniformly ; and from it we may infer that both must be propa- 

 gated uniformly at the same time. To the first order of approxi- 

 mation, V = fl'c\ Applying this relation in the problem before 

 us, it having been shown that V is propagated with the uniform 

 velocity ica, we have 



Y=/cacr = ~F(fcat— z + c). 



36. If the propagation had taken place in the opposite direc- 

 tion, we should have similarly obtained 



V' = — Kao* =f(/cat + z + d) . 



As the equations from which these results are deduced are linear 

 with constant coefficients, motions propagated in opposite direc- 

 tions may coexist. Hence putting Y 1 for V + V, and <r x for 

 <7-f o 7 , we shall have generally, 



Y l ='¥{icat— z + c)-\-f(/eat+z + c/), 



a 1 =Y(Kat—z + c)—f(/cat + z + c'). 



From these equations we obtain by differentiation, 



This equation is applicable generally to motion in parallel straight 

 lines. Now when the motion is such, a prismatic portion of the 

 fluid of indefinitely small transverse section may be conceived to 

 be separated from the rest, and to be contained within rigid 

 boundaries. The above equation will thus apply to constrained 

 rectilinear motion ; and it may be remarked that it differs from 

 the analogous equation applicable to free motion by having the 

 factor /c 2 in the first term. It should also be noticed that the 

 composite character of the motion is a physical reality in con- 

 strained rectilinear motion. 



37. Although in the investigation of equation (15) the motion 

 was supposed to be in parallel straight lines, that equation is 

 equally applicable to motion within slender tubes of variable 

 transverse section, if the inclinations of the surface to the axis be 

 everywhere indefinitely small. For the factor /c 2 simply takes 

 account of the effect of preventing transverse motion ; and as 



