the Vibrations of an Elastic Fluid. 143 



the time t be a 2 (l + a), cr being a small quantity the powers of 

 which above the first are neglected. Then, by known equations, 



a?da du a 1 da dv _ a 2 da dw n 



dec dt ' dy dt ' dz dt 



From these equations it is easily seen that 



\d 2 u d% n d 2 u d 2 w d% d^w n 



dt dy dtdx ' dtdz dtdx 3 dtdz dt dy 



Hence by integration, 



du dv p du dw _ ~ f dv dw „ n 



dy dx ' dz dx ' dz dy" } 



the functions C, C, C" being independent of the time. But 

 from what has already been ascertained respecting the motion 

 which is the subject of this investigation, we may infer that it is 

 vibratory, so that neither u, v, iv nor their partial differential 

 coefficients with respect to coordinates can be independent 

 of the time. Hence C=0, C' = 0, C" = 0, and consequently 

 udx + vdy + wdz is an exact differential. The same proposition 

 is proved in a somewhat different manner in the communication 

 of December 1852. 



26. This theorem being established, it will not be necessary, 

 after assuming udx + vdy + wdz to be an exact differential, to 

 restrict the reasoning to positions immediately contiguous to the 

 axis of rectilinear motion. Having regard, however, to the an- 

 tecedent general demonstration of the existence of such an axis, 

 we shall still assume that 



(d . f(j>) = udx + vdy + wdz, 



f being a function of x and y, and (/> a function of z and t. The 

 legitimacy of the assumption will appear in the sequel of the 

 reasoning. Thus we have, as before, 



, df , df .deb 



"=*» V= *Iy' W= fTz- 

 Also the equation which gives the condensation a to the first 

 order of approximation is 



the arbitrary function of the time being caused to vanish on the 

 principle, already adopted, of conducting the reasoning indepen- 

 dently of any arbitrary conditions. By eliminating a between 

 this equation and the second general equation taken to the first 

 order of approximation, viz. 



da du dv dw 



dt dx dy dz" ? 



