54- Mr. G. G. Stokes 07i some Points in the 



kindly explained to me his reasoning, which he has since 

 stated in detail (vol. xxxiii. p. 463). 



The whole force of the reasoning rests on the tacit suppo- 

 sition that when a wave is propagated from the centre out- 

 wards., any arbitrary portion of the wave, bounded by spherical 

 surfaces concentric with the bounding surfaces of the wave, 

 may be isolated, the rest of the wave being replaced by qui- 

 escent fluid; and that beingsoisolated, it will continue tobepro- 

 pagated outwards as before, all the fluid except the successive 

 portions which form the wave in its successive positions being 

 at rest. At first sight it might seem as if this assumption were 

 merely an application of the principle of the coexistence of 

 small motions, but it is in reality extremely different. The 

 equations are competent to decide whether the isolation be 

 possible or not. The subject may be considered in different 

 ways; they will all be found to lead to the same result. 



1. We may evidently without absurdity conceive an out- 

 ward travelling wave to exist already, without entering into 

 the question of its original generation ; and we may suppose 

 the condensation to be given arbitrarily throughout this wave. 

 By an outward travelling wave, I mean one for which the 

 quantity usually denoted by <p contains a function of r — at, 

 unaccompanied by a function of r + «/, in which case the ex- 

 pressions for V and s will likewise contain functions of 7 — at 

 only. Let 



flr^— •£ ' (1.) 



r 



We are at liberty to supposey'(2;) = 0, except from z = b to 



2 = c, where b and c are supposed positive; and we may take 



,f'{z) to denote any arbitrary function for which the portion 



from z = b to z=c has been isolated, the rest having been 



suppressed. Equation (1.) gives 



(p=-a^J"sdt='&^-=^+^r), . . . (2.) 



\I/(r) being an arbitrary function of r, to determine which we 

 must substitute the value of <p given by (2.) in the equation 

 which <p has to satisfy, namely 



^^^...^t. ..... (3.) 



This equation gives \|/(r) = C H , C and D being arbitrary 



constants, whence 



^^^f^ f'{r-at) Ar-at) _D ^ ^ ,^, 

 dr r r^ ^2* • • v ; 



