the Radiation of Heat on the Propagation of Sound. 311 



increase in the force of restitution of the piston, arising from the 

 alternate elevation and depression of temperature, is analogous 

 to the increase in the forces of restitution of the particles of air 

 arising from the same cause, to which corresponds an increase 

 in the velocity of propagation of sound. 



Another consequence follows from the formula (13.), which 

 deserves to be noticed. We have already seen that this formula 

 gives n/A-' sec y}r for the value of V, the velocity of propagation. 

 Putting for shortness 



l+a;8 = K, (14.) 



we get from (11.) and (12.), 



,^ 2^(KV+g^) 



Kn^ + q^+^/{(K^n'' + g^){n^ + q^)}• • ' ^^^■> 



Hence if 5- be comparable with n, V, which is a function of the 

 ratio of q to n, will change wiih. n, and therefore the velocity of 

 propagation will depend upon the pitch, which is contraiy to 

 observation. But if q be either incomparably greater or incom- 

 parably smaller_than n, V will assume one or other of its limiting 

 values \/k, x^kK ; and the velocity of propagation will be inde- 

 pendent of the pitch, as observation shows it to be. We are thus 

 led, by considering the velocity of propagation, to the same con- 

 clusion as was deduced from the circumstance that sound is 

 capable of travelling to a distance. 



Since, then, we are diiven to one or other of the alternatives 

 above mentioned, it only remains to decide which we must choose. 

 But before entering on this subject, it will be proper to consider 

 whether the fonnula (13.) is of sufficient generality. 



In the first place we may observe, that the formula (13.) is 

 only a particular integral of (7.). It is adapted to the case in 

 which the motion is kept up by a vibrating plane, which agrees 

 rao.st nearly with the circumstances of ordinaiy experiments; 

 but a particular law of disturbance as regards the time is assumed, 

 namely, that expressed by a single circular function. Now we 

 know that any periodic function of the time, having t for its 

 period, may be expressed by the sum of a finite or infinite nmnber 

 of circular functions having for their periods t and its submul- 

 tiple.s ; and even a non-periodic function may be expressed by a 

 definite integral, of which each element denotes a circular func- 

 tion. So far, therefore, the formula (13.) is of sufficient gene- 

 rality. 



In the next place, the fonnula (13.) applies to motion in one 

 dinicnsioii only. But had we employed the general equations 

 (1.), (2.), wliicli relate to motion in three dimensions, we should 

 liave obtained the same partial diffi-rential equation as (7.), with 



