1 18 Mathematical Theory of the Vibrations of an Elastic Fluid. 



propose to advert to some remaining points of this mathematical 

 theory which may require elucidation, and to draw some infer- 

 ences from the foregoing- results, particularly with reference to 

 the undulatory theory of light. 



Before concluding this communication, I beg to advert to the 

 reference made by Prof. Tyndall, in the Philosophical Magazine 

 for last April, to the discussion which, as he says (§ 2), was 

 "carried on in 1851 between Professors Challis and Stokes 

 regarding Laplace's correction for the theoretic velocity of 

 sound/'' From his expressing a hope that I shall deem con- 

 clusive the results of the experiments on radiant heat to which 

 he refers, I hardly think he can be fully aware of the points in- 

 volved in that discussion. The results of experiments made by so 

 able a physicist I accept as valuable and trustworthy; but I fail 

 to see how those to which he directs my attention settle the 

 question about the velocity of sound, the precise nature of which 

 will perhaps be understood by the following statement. Laplace, 

 who, it is well known, did not give much attention to the phy- 

 sical applications of partial differential equations, was content to 

 deduce from hydrodynamics the value of the velocity of sound 

 first obtained by Newton. In this he has been followed by later 

 mathematicians, as Poisson and others, who, in deducing the 

 rate of propagation, have adopted a process first suggested by 

 Lagrange. After much consideration given for a long time to 

 the mathematical theory of hydrodynamics, I discovered that 

 that process is objectionable in principle, and that by the appli- 

 cation of correct mathematical principles (which are fully ex- 

 plained in this communication) a value of the velocity is obtained 

 so accordant with that given by observation as to leave scarcely 

 any residual quantity to account for. This is the real question 

 at issue between me and Prof. Stokes, who contends for the 

 Newtonian value as ordinarily deduced ; and it will thus be seen 

 that, antecedently to any physical considerations, there is a 

 question to be settled which is exclusively a mathematical one. 



With respect to the attempt made by Laplace to account for 

 the difference between the Newtonian and the observed velocities 

 of sound, I fully believe that such a theory as that which he 

 proposed would never have been thought of if the principles of 

 the application of partial differential equations to the determina- 

 tion of the motion of fluids had been better understood. Laplace's 

 physical theory has not been admitted in all particulars ; but the 

 general conception that the heat developed by sudden condensa- 

 tion, or absorbed by sudden rarefaction, produces the excess of 

 velocity, is accepted. To make, however, this general view bear 

 upon the fact, three distinct hypotheses have been made : — 

 (1) that a certain amount of increment of temperature always. 



