90 Prof. Challis on the t^rincipLes of Hydrodynamics. 



Consequently, by substituting the above value of e, 



Velocity of propagation —a a/ 1 + -j. 



This value accords with the experimental determination of the 



velocity of sound. (See Phil. Mag. vol. xxxiv. p. 98.) 



For the purpose of substantiating the above reasoning, I pro- 

 ceed to notice two objections which may be expected to be urged 

 against it ; one drawn from analytical, and the other from phy- 

 sical considerations. First, it may be said, that when r is inde- 

 finitely large, the middle term of the equation 



is incomparably less than the other two, and that the value of/ 

 is very approximately given by the solution of the equation 



viz. /=Ci cos2 ^/^(r + Cg). According to this reasoning, the 

 consecutive large values of r which cause / to vanish increase by 



IT 1 



the common difference - — -r., and not by -—^. To this argu- 



ment I reply, that the middle term of the above equation is not 

 small compared with the others for every large value of r, in 

 proof of which it is only necessary to remark, that for all the 



maximum values of that term -rs- is equal to -~ . The form of 



dr^ rdr 



the function or series for / depends therefore essentially on that 

 middle term, however large r be taken, and is not even approxi- 

 mately obtained if that term be omitted. 



Again, it may be urged that the excess of the observed velo- 

 city of sound above the value a is physically accounted for by the 

 generation or absorption of heat consequent upon a sudden con- 

 densation or rarefaction of the air. It is proved by experiments 

 on air contained in closed spaces, that a sudden condensation 

 raises the temperature and a sudden rarefaction depresses it. 

 But such experiments seem to be inapplicable to the case of 

 sudden changes of density in air which is not enclosed, and 

 cannot be safely depended upon for the determination in that 

 case of the effect on the temperature and elasticity of the air. I 

 am not aware of any experiment having been made by which 

 any change of temperature or elasticity has been shown to accom- 

 pany sudden changes of density in free air. And even if it be 

 admitted that a change of elasticity is produced in that manner, 

 the observed velocity of sound is not accounted for without 



