the Velocity of Sound. 277 



The last of these equations gives, by means of the other three, 

 d^s „(d^s , dH dh\ ^ 



Suppose, for the moment, that s has been obtained from this 

 equation by integration. Then for the velocities we have, 



n 2 r^^^ u ry a d.fsdt 



J dx dx 



w = C" - a^ r~ dt = C" - a^. iZ^^, 

 J dz fj^ 



where C, C, and C" are functions of co-ordinates only. It 

 is to be observetl that these values of u, v, xa are perfectly 

 general, being obtained prior to any consideration of the way 

 in which the fluid is put in motion, and consequently apply to 

 all points of the fluid in every instance of motion in which 

 powers of the velocity and condensation above the first may 

 be neglected. Now the motions we are about to consider are 

 vibratory, or at least, not such that any part of the velocity 

 remains permanently the same at the same point of space for 

 any length of time. Consequently C = 0, C' = 0, and C" = 0. 

 Hence if \I/= —a^Tsdl, 



c/vl/ (Zv|/ d-h 



"=rf^' '^=^' ^=5^' 

 and 



udx + vdy + "wdz = {(Hi)^ 



an exact differential. 



It is thus shown that the condition that udx + vdy + iadz be 

 an exact differential, must be satisfied in a manner that shall 

 equally apply whatever be the original disturbance of the fluid. 

 The supposition that -^ is the product of two functions ip and /, 

 such that (p does not contain x and j/, and /does not contain 

 z and /, will be shown in the sequel to iulfil this requisite. 

 On this supposition, 



df df .d<p 



«=<P-r-5 v=<p^, '!3y=/-r> 

 ^ dx ^ dy "^ dz 



and 



udx + vdy + 'wdz = (p (^£ dx+'f^ dy) +/-£ dz, 



