498 The Rev. J. Challis on the Velocity of Sound. 



and a function of 2 and t\ according to which, if h^ have a 

 value different from zero, the waves are neither plane nor 

 spherical. No incompatibility here pi'esents itself, at least, as 

 far as to the first order of approximation. We may even ex- 

 plain, on this hypothesis, the contradictions met with in the 

 other two. For supposing non-divergence to be the normal 

 condition of waves, a plane-wave, or a spherical wave, is pos- 

 sible only when it is compounded of an unlimited number of 

 non-diverging waves : and in the case of a spherical wave so 

 compounded, the variation of condensation is inversely as the 

 square of the distance from the centre, as we have shown that 

 it ought to be. 



The argument respecting this third supposition is evidently 

 incomplete, unless it be proved that no incompatibility is met 

 with when the reasoning is conducted by exact equations. I 

 propose in another communication to consider this part of 

 the subject, which requires a mode of treatment different from 

 that in my paper on Luminous Rays (Camb. Phil. Trans., 

 vol. viii. part 3, p. 365). I have there made use of equation 

 (A.), not being at the time aware of what is the true interpre- 

 tation of its integral. Consequently the equation which Mr. 

 Airy has objected to, though not upon a consideration of its 

 merits, and which I have avoided introducing into this com- 

 munication, is as yet unsupported. 



What I have said above will suffice for an answer to Mr. 

 Airy's arguments in paragraphs (/S.) and (y.). In the former, 

 the motion results from two systems of plane- waves propagated 



e 



n 



in directions making an angle whose cosine is with the 



e 



« + - 



axis of ^ on opposite sides, and therefore diverging from that 

 axis. In the latter, the motion results from systems of plane- 

 waves also. I have nothino; to urfre against the mathematical 

 reasoning : my sole answer is, that plane-waves are physically 

 impossible. 



The difficulty respecting the augmentation of the velocity 

 of sound by the development of heat, cannot be so summarily 

 disposed of as Mr. Airy appears to imagine. I shall perhaps 

 succeed better in conveying my meaning by using symbols. 

 If fl be the temperature where the pressure is j) and density p, 

 and 9| the temperature in the quiescent state of the fluid, we 

 have, by a known equation, 



