the Radiation of Heal on tkt Propagath'n tif Sound. 313 



root of the period. In the preseAt case, the period with which 

 we have to deal is that of a sonorous vibration/ a time which 

 must be regarded as extremely small in questions relating to the 

 conduction of heat. Hence, if r be the period of vibration, the 

 fluctuation of temperature of the tube will be a small quantity 

 of the order a/t compared with the flux of heat. Now if H, h 

 be the interior and exterior conductivities, v a normal to the inner 

 surface of the tube, drawn from the tube inwards, 6' the excess 

 of temperature of the tube above the mean temperature 0^; and 

 'if we suppose the surface to be plane, and to radiate into an in- 

 finitely extended medium at a temperature 6^ + 6, where 6 is 

 supposed to be constant as regai'ds space, but to be a periodic 

 ^function of the time, w§ must have at the inner surface of the 

 'tube ' ^ .i'..,i - : .._.._ ;: _::j ■; 



^disii ^Borjd;;ol, bIiAMrfi7.i,?^"-^l=9;,„u.x -tn ...„^>. •,.„(.,. 

 Now. according to what has b^li already remarked, ^ is a small 



gqnautity of tl^. order fV^r compfired with -7- ; and it follows from 



ilie above equation, that -7- is comparable with 6, and therefore 



6' is a small quantity of the order Vt compared with 9. Hence, 

 i'efven in the case above supposed, the fluctuation of temperature 

 ybf the tube at the surface woidd be very small. But in the 

 ''Actual case, the tube radiates, not into an infinite medium, but 

 merely across the air contained within it, beyond which is situated 

 the opposite face of the tube, at a temperature equal to the first 

 face ; and therefore the fluctuation of temperature of the inner 

 surface of the tube will be far smaller than in the case supposed 

 above, so as to be altogether insensible. Hence the air radiates 

 within an envelope at a temperature ^q, so that the radiation 

 takes place as if the air heated by compression radiated into an 

 infinite medium at a temperature 6q. Of com'se the same rea- 

 soning will apply to the apparent radiation of cold. Hence the 

 formula (13.) may be applied without change to the vibration of 

 the air within a long tube, and accordingly may be employed in 

 considering the experiments of M. Biot above alluded to. 



The preceding view of the eff"ect of radiation within a tube is 

 veiy ditterent from that taken by M. Poisson in his Traite de 

 Mecaaique (vol. ii. art. 6(io). The latter, however, is contained 

 in a mere passing remark oft'cred by way of conjecture, and pro- 

 bably written without much consideration, and therefore ought 

 hardly to be regarded as supported by Boisson^s authority. 



lyct us now ])iiss to numerical values, in order to make out, 

 independently of any* assumption respecting the true exjjlanation 

 of the velocity of sound, whether <] must be regarded as very 

 Phil. May. S. 4. Vol. 1 . No. 4. April 1 H5 1 . Y 



