the Radiation of Heat on the Propagation of Sound. 309 



It will be easily seen that the expressions for 6, u, and p are 

 of the same form, that is to say, that they involve the same ex- 

 ponential multiphecl by a sine or cosine of the same angle. Had 

 the actual expressions been required, it would have been shorter 

 to defer the suTjstitution of real for imaginary quantities until 

 after the imaginary expressions for 9, ii, and p had been obtained. 



Now the formula (13.) shows, that unless sin ■v/r be insensible, 

 soimd cannot be propagated to a distance, but must be stifled 

 in the neighbourhood of the vibrating body by which it is excited. 

 Since we know very well that this is not the case, we are taught 

 that sin -v/r is insensible, and therefore -^ itself, since ^Ir denotes 



an angle lying between and — . The formula (13.) shows, that 



if V be the velocity of propagation, V = ?i/i~^ sec -s^, which, when 

 -<|r is insensible, reduces itself to njx'K Referring to (12.), we see 

 that, in order that i|r may be insensible, it is necessaiy to sup- 

 pose, either that q is incomparably greater than n, or that n is 

 incomparably greater than q. On the former supposition the 

 formula (11.) gives V= \^k, which is equival ent to Ne wton^s 

 result. On the latter supposition we get V= */h{\ + a.^), which 

 is equivalent to Laplace's result. 



The reason why sound would be so rapidly stifled were q and 

 n comparable with each other, may be easily seen on taking a 

 common-sense view of the subject. Conceive a mass of air con- 

 tained in a cylinder in which an air-tight piston fits, which is 

 capable of moving without friction, and which has its outer face 

 exposed to a constant atmospheric pressru'c; and suppose the 

 air alternately compressed and rarefied by the motion of the 

 piston. If the motion take place with extreme slowness, there 

 will be no sensible change of temperature, and therefore the 

 work done on the air during compression will be given ou.t again 

 by the air during expansion, inasmuch as the pressure on the 

 piston will be the same when the piston is at the same point of 

 the cylinder, whether it be moving forwards or back^vards. 

 Similarly, the work done in rarefying the air will be given out 

 again by the atmosphere as the piston returns towards its posi- 

 tion of equihbrium, so that the motion would go on without any 

 permanent consumption of labouring force. Next, suppose the 

 motum of the jjiston somewhat quicker, so that there is a sen- 

 sible change of temperature produced hy condensation and rare- 

 faction. As the piston moves forward in condensing the air, the 

 teiJiperature rises, and therefore the ])istou has to work against 

 a pressure greater than if there had been no variation of tempe- 

 rature. ]iy the time tlic ])iston returns, a good portion of the 

 heat devehjped by compression has passed oft^ and therefore the 

 piston is not helped as mucli in its backward motion by the pres- 



