the Radiation of Heat on the Propapation of Sound. 307 

 5 and 6, which are both small quantities of the first order, we get 



jj = kp{l + s + ocd} (3.) 



It remains to form the equation relating to the changes of 

 temperature. Let ^s be the eleration of temperature produced 

 by a sudden small condensation s. The condensation which a 

 given element of the fluid receives in the time dt is equal to s'dt, 

 where 



, ds ^ ds ds ds ds 



'=dt+^d^+''Ty+"'d-. = dt''''''^y' 

 and the elevation of temperature due to this condensation is equal 

 to 0s'dt. We know that heat radiates freely to great distances in 

 air, and therefore, of the heat which radiates from the element 

 considered, we may neglect the small portion which may be 

 absorbed by the air in its neighbom-hood, and consider only what 

 goes to great distances. Hence the result will be sensibly the same 

 as if the element radiated into a medium having the constant tem- 

 perature Oq, which is the mean temperature of the whole. The 

 quantity, then, which escapes from the element during the time 

 dt, will be proportional to the small excess of the temperature 

 of the element over the mean temperature of the medium ; and 

 the consequent depression of temperatm-e may be expressed by 

 qOdt, where g is a constant which may be called the velocitij of 

 cooling referred to a difference of temperature unity. We have, 

 therefore, 



d6 -ds n 



Tt=^dt-'i^ (4-) 



The SIX general equations (1.), (2.), (3.), (4.) serve, along with 

 the equations of condition relating to any particular problem, to 

 make known the six unknown quantities u, v, w, p, s, 6. 



To simplit^^^ the question as much as possible, I shall take the 

 case of plane waves. Taking the axis of x pei-]jendicular to the 

 planes of the waves, we have w = 0, «; = 0, and u, p, s, d will be 

 functions of only x and t. The equations (1.) and (2.) become 



dp du ds du 



^ = -^^' dt'^ll=^' • • • • (5.) 



and eliminating/? and u from these equations and (3.), we get 



dh ,(dh dW\ 



rf^=H^+«^^; (6-) 



Eliminating 6 between (4.) and (0.), we get 



