24 Prof. Challis on the Central Motion of an Elastic Fluid, 



and at the same time its condensation vary inversely as the 

 distance ; whereas it is certain that in this case the condensation 

 must vary inversely as the square of the distance in order that 

 the quantity of the fluid may remain constant. The former in- 

 ference is, however, strictly drawn according to the principles 

 usually adopted in the application of analysis to the motion of 

 fluids, the very same process being employed in determining the 

 velocity of propagation of vibrations. 1 do not therefore see 

 how, on those principles, the difficulty I have indicated can be 

 overcome. What would be at once said according to the views 

 that I take is, that all such reasoning is essentially faulty, 

 because circumstances of the motion which are not arbitrary are 

 thus made to depend on the arbitrariness of the functions. My 

 reason for adverting to this subject again is, that I have recently 

 found, contrary to what I had supposed to be the case, that the 

 law of the variation of condensation according to the simple in- 

 verse of the distance, is in strict accordance with the views that 

 I advocate, and consistent with the principle of the constancy 

 of mass. These results have been arrived at in the following 

 manner. 



At the end of a communication to the Number of the Philo- 

 sophical Magazine for February 1853, I have given a solution 

 of the problem of the central motion of an elastic fluid, which, 

 although not as general as it might be made, conforms exactly 

 to the principles antecedently laid down. The solution proceeds 

 on the hypothesis that the central motion results from the com- 

 position of an unlimited number of the simple vibratory motions, 

 w^hich, as already stated, are deducible from the hydrodynamical 

 equations prior to any case of arbitrary disturbance, the axes of 

 the motions passing equally in all directions through the centre. 

 The central motion and law of condensation are thus obtained 

 independently of any case of disturbance, and must consequently 

 be such only as result from the mutual action of the parts of the 

 fluid. The expressions for the velocity and condensation which 

 this process gives are the following : — 



v= — • •< cos— -(r — Kof + c)-i- cos— -{r + Kat + c) >■ 

 ?• L A- A. J 



— ^^' •{ sin-:- {r — Kat + c) + sm%--{r + Kat + c) > 



4771' LA, A, J > 



m f 27r , . , X Stt . , , \ \ 



Kns=-- -{ cos — {r — Kat+cj — cos-— {7- + Kaf + c) >. 



If we consider apart the terms relating to propagation from the 

 centre, it will appear from the term — ^ — ^r sin— {r—Kat + c), 



