306 Prof. Stokes's Easamination of the possible effect of 



the velocity of sounds as the development of lieat by sudden con- 

 densation. 



Professor Potter's objection to the received theory, on the 

 ground that the velocity of sound is independent of its loudness 

 and pitch, is, in fact, no objection at all, inasmuch as this is a 

 direct consequence of the theory in question. 



Inasmuch as Laplace's foraiula is a rigorovis deduction from 

 the physical hy])otheses adopted, there is no way of escaping 

 from his result but by calling in question the hypotheses them- 

 selves. Now the development of heat and cold by sudden con- 

 densation and rarefaction is not merely a hyjjothetical cause, the 

 only evidence of whose existence is that it explains the phaeno- 

 mena, but is a well-known physical fact, proved by direct expe- 

 riment. That in the case of small sudden condensations (positive 

 or negative) the increase of temperature is ultimately propor- 

 tional, ceteris paribus, to the condensation, will not, it is pre- 

 sumed, be called in question. The only way, then, of escaping 

 from the conclusion that the velocity of sound is really increased 

 by the cause assigned is, to suppose that the heat produced by 

 condensation passes away so rapidly by radiation that the result 

 is the same as though condensation and rarefaction were inca- 

 pable of changing the temperature of air. This supposition has, 

 in fact, already been made by Professor Challis. The main 

 object of the present communication is to examine the con- 

 sequences of such a supposition, in order to make out whether 

 it be tenable or not. 



Let us take the case of an infinite mass of homogeneous elastic 

 fluid, acted on by no external forces, and having throughout a 

 uniform temperature, and consequently a imiform pressure, ex- 

 cept in so far as the pressure, and consequently the temperature, 

 are aflfected by small vibratory movements. Let the fluid be 

 referred to the rectangular axes of x, y, z ; let u, v, w be the 

 components of the velocity, /the time, p the pressui-e, p the 

 density in equilibrium, p[\+s) the actual densitj^, so that * is 

 the condensation. The three ordinary equations of motion and 

 the equation of continuity become in this case, on neglecting as 

 usual the squares of small quantities, 



dp _ du dp _ dv dp _ dw 



d^~~Pdt' diJ~~^dF' Ts~~~Pli' ' ' ^' 



ds du dv div _ ,^ . 



dt dx dy dz 



Let 6^ be the temperature in equilibrium, d^^Q the actual 

 temperature. Then ;? = A-Qp(l + s) (1 + ol^Oq -f Q) . Putting k for 

 ^o(l + ao^o)^ '*■ for ao(l + «o^(,)~', and neglecting the product of 



