received Theory of Sound. 59 



object for which the problem was investigated. Since there 

 is no isolated wave of condensation travelling outwards, the 

 complete solution of the problem leads to no contradiction, 

 as might have been confidently anticipated. 



How then stands the theory of sound as usually received? 

 So long as we confine our attention to the first order of small 

 quantitiep, which is a perfectly legitimate mode of approxima- 

 tion, there is neither contradiction nor difficulty ; for Professor 

 Challis's difficulty with respect to the effect of the develop- 

 ment of heat by sudden compression, in the altered form in 

 which he has now put it, has nothing to do with the first order 

 of small quantities. On employing exact equations, it is true 

 that a very remarkable kind of motion has been brought to 

 light in the course of the discussion, and shown to be possible, 

 if not in air, at least in an ideal fluid in which the pressure is 

 equal in all directions, and varies as the density. The pre- 

 cise nature of this motion I do not pretend to describe. I 

 have already stated (vol. xxxiii. p. 352) that I had grounds 

 for believing that a sort of reflexion would take place; though 

 whether this reflexion would or would not prevent the forma- 

 tion of what I have called a surface of discontinuity I am un- 

 able to say, although I am inclined to think that it would. 

 , To prevent misapprehension, I will observe that it is the mo- 

 tion, whatever be the nature of it, which takes place after the 

 quantity denoted at page 351 by A becomes infinite, that I 

 have referred to in using the word bore : I did not mean, in 

 using that term, to assert that a surface of discontinuity was 

 certainly formed. An interesting field of inquiry lies open 

 with reference to the possibility of an actual approximation to 

 a bore in the case of fluids such as they exist in nature, and 

 generally with reference to the modification of sound by di- 

 stance. It is to such an inquiry that the consideration of the 

 effect of different functional relations between p and /?, when 

 the changes of p are too great to be considered proportional 

 to those of ^, properly belongs, and in particular the functional 

 relation which connects p and p in the case of air, in conse- 

 quence of the heat developed by sudden compression. But 

 as regards the purely mathematical question of the treatment 

 and interpretation of our equations, no contradiction arises 

 when the restrictions which the occurrence of infinite quanti- 

 ties imposes on all mathematical reasoning are attended to. 



In conformity with an intention which I have already ex- 

 pressed (vol. xxxiii. p. 349), and with the title which I have 

 chosen for this article, I have confined myself to the defence 

 of the theory of sound as usually received. I have refrained 

 from calling in question the new and startling conclusions 



