Prof. Challis on the Principles of Hydrodynamics. 93 



element between the sections m and m'. That is, neglecting 

 terms involving ^t Bz and Bz'^, let 



mpYBt — m'p'VSt = mpBz — m'p'Bz. 

 Then the mean density of the former element will have been 



Sz 

 transferred through the space Sz in the time of Bt, and ^ is its 



rate of propagation, which ultimately applies to the density p. 

 Hence by the above equation, passing from differences to dif- 

 ferentials, 



d . mpY ___ d.mp ^ Bz_ 



dz ~ dz Bt' 

 Now as m is of arbitrary value, the rate of propagation cannot 

 be assumed to be uniform excepting so far as the changes of p 

 and V are independent of the changes of m. Omitting, therefore, 

 the differentiation with respect to m in the above equation, and 

 supposing the rate of uniform propagation to be «', we have 

 d^^^, dp 

 dz ' dz* 



and by integration, 



the arbitrary function being a function of m as well as t, because 

 the differentiation with respect to m was omitted. For small 

 motions, putting 1 + cr for p and neglecting the product cV, 



V=«'(7 + x(m,0. 

 The effect of the variation of m is taken into account by making 

 a given phase of the condensation vary inversely as the transverse 

 section of the tube. Hence if yjr{z) be the transverse section at 

 any distance z from the origin, the nature of the supposed kind 

 of motion will be expressed generally by the equations 



It is to be remarked, that if %(m, ^) =0, the resulting equation 

 between V and a, viz. Y=a'o; is different from that obtained for 

 the case of unconstrained motion along an axis of propagation. 

 In the latter case we had KW—Kacr, Ka being the rate of propa- 

 gation. The reason of the difference appears to be, that the trans- 

 verse vibrations which take place in free motion are destroyed in 

 constrained motion by an impressed transverse force, which alters 

 the relation between the velocity and condensation without alter- 

 ing the rate of propagation. 



Recurring now to the general equation 



dp d.Yp ^^ f\ \\ ^ 



