Prof. Challis on the Hydro dynamical Theory of Vibrations. 293 



term being arbitrary, it follows from the preceding reasoning 

 that these values of ¥(/cat) are propagated with the velocity ica } 

 and that for any distance z from the origin we have 



Y=~F{fcat—z + z l ). 



It is here to be remarked that there is no reason from the fore- 

 going argument to conclude that this value of V should satisfy 

 the equation (6) when any arbitrary form is given to the func- 

 tion F, inasmuch as that equation applies only to motion along 

 a single axis, accompanied by transverse motion. 



35. We have now to find the relation between the velocity and 

 condensation in the case under consideration, of motion in 

 parallel straight lines. The relation cannot be the same as that 

 obtained in art. 33 for motion along an axis, because the con- 

 densation on the axis must be affected as well by the transverse 

 as by the longitudinal vibrations. Since, from what is shown 

 above, the rate of propagation of the velocity is constant, it may 

 be proved by reasoning given under Prop. XII. in the Number 

 of the Philosophical Magazine for February 1853, that a neces- 

 sary relation exists between the velocity and the corresponding 

 density. For the sake of distinctness I shall here repeat so 

 much of that reasoning as is required for completing the present 

 argument. Let V and p be the velocity and density of the fluid 

 which passes the transverse section m of a slender straight tube 

 at the distance z from the origin at the time t, and let V and p 1 

 be the velocity and density of the fluid which at the same 

 instant passes the transverse section m 1 situated in advance of the 

 other by the interval Sz. The increment of fluid between the two 



sections in the interval from t — — to t-\- jr ismpV8t—m!p f Y l Bt 3 



because the changes of pV and p'V 7 in the small interval St may 

 be supposed to be proportional to that interval, and may therefore 

 be omitted. Let the above increment become equal to the excess 



of the quantity of fluid at the time t =- in the element of 



length Sz terminating at the section m, above the quantity of 

 fluid at the same time in the element between the sections m 

 and m'. That is, neglecting terms involving SzSt and Sz 2 , let 



mpYBt — m'p 1 V'St = mpSz — m'p'Sz. 

 Then the mean density of the former element will have been 



Sz . 

 transferred through the space Sz in the time St ; and -k- is its rate 



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

 by the above equation, passing from differences to differentials, 



d . mpV d . mp Sz 



dz dz St 



