GENEKAL THEOEY OF SOUND WAVES 201 



If A denote the dilatation of volume of the fluid which at 

 the instant t fills the space SasSySz, as compared with its 

 equilibrium condition, we evidently have 



-^- = div (u, v, w\ (5) 



or since, in the case of small motions, s = A, 



The equations (4), (6) are fundamental in the present branch of 

 our subject. The purely kinematical relation (6) is sometimes 

 called the " equation of continuity." 



69. Velocity-Potential. 



If we integrate the equations (4) of 68 with respect to t 

 we obtain 



,1 



OX j || v_i/ j u i / 1 \ 



where U Q , v 0t W Q are the values of u, v, w at the point (a?, y, z) at 

 the instant t = 0. In a large class of cases, these initial values 

 of u, v, w can be expressed as the partial differential coefficients 

 of a single-valued function of (x, y, z), thus 



Throughout any region to which this statement applies, the 

 values of u, v, w at any subsequent instant t can be similarly 

 expressed; thus, from (1), 



- ' ......... 



rt 

 where <^> = c 2 I sdt+fa ................... (4) 



J o 



This function <f> is called a " velocity-potential," owing to its 

 analogy with the potential-function which occurs in the theories 

 of Attractions, Electrostatics, &c. It was introduced into 

 hydrodynamics by Lagrange. 



