263-264] SPHERICAL WAVES. 479 



inversely as the distance from the origin. The velocity due to 

 the same train of waves is 



As r increases the second term becomes less and less important 

 compared with the first, so that ultimately the velocity is pro 

 pagated according to the same law as the condensation. 



264. The determination of the functions F and / in terms of 

 the initial conditions, for an unlimited space, can be effected 

 as follows. Let us suppose that the initial distributions of velocity 

 and condensation are determined by the formulae 



where i/r, % are arbitrary functions, of which the former must 

 fulfil the condition ^r (0) = 0, since otherwise the equation of 

 continuity would not be satisfied at the origin. Both functions 

 are given, primd facie, only for positive values of the variable ; 

 but all our equations are consistent with the view that r changes 

 sign as the point to which it refers passes through the origin. On 

 this understanding we have, on account of the symmetry of the 

 circumstances with respect to the origin, 



+ (-r) = + (r), x(-r) = x (r) ............... (8), 



that is, -\|r and % are even functions. From (6) and (7) we have 



If we put 



J%X(r)&amp;lt;fr = Xl (r) ..................... (10), 



the latter equation may be written 



= i Xl (r) .................. (11), 



the constant of integration being omitted, as it will disappear 

 from the final result. We notice that 



%i(- )=%0) ..................... (12). 



