hy Distance of Propagation, 24S 



meaning can be given to the solution up to a certain point in 

 the progress of the wave." If we may believe these two ma- 

 thematicians, the wave at this point suddenly emancipates 

 itself from the control of analysis, and the subsequent motion 

 can be pursued by no other means than conjecture. Accord- 

 ingly each has favoured us with his conjectures. Mr. Airy's 

 are contained in the article which has given occasion to these 

 remarks, and Mr. Stokes's in the communication already re- 

 ferred to (Phil. Mag. vol. xxxiii. pp. 352-356). 



It is not my intention to enter into any consideration of 

 these conjectures, which, in my opinion, can only be regarded 

 as proofs that the authors of them have fallen into an erro- 

 neous course of reasoning. In any legitimate application of 

 analytical calculation to a physical question, a spontaneous 

 failure of the analysis, leaving no resource but conjecture, can 

 never occur. Before such necessity has arisen, some fault in 

 logic must have been committed. I do not doubt that I point 

 out the fault committed in the case under discussion by saying, 

 that certain consequences deduced by analysis from the pre- 

 mises are received, while other consequences equally deduced 

 by analysis from the same premises are rejected. The ana- 

 lytical solution of the problem of plane-waves is, as I have 

 already stated, that the velocity xi at any point of the wave 

 and the corresponding density are propagated with the velo- 

 city a-\-v. In consequence of the different rate of propagation 

 of different parts of the wave, two different states of the me- 

 dium must occupy eventually the same position at the same 

 time. For instance, a position of maximum velocity is a po- 

 sition of no velocity when the interval t has become , — ; that 



is, taking the numerical example before adduced, after the 

 lapse of two seconds and a half, when the wave has travelled 

 through 2290 feet. We are thus brought by the analysis to 

 an absurdity. It is true the absurdity is first consummated 



at the end of the interval - — . Mr. Stokes avails himself of 



this circumstance to limit his acceptance of the results of 

 analysis to this interval, and avoids the consideration of the 

 absurdity ; whereas right reasoning demands, if in any case 

 an absurdity is strictly deduced from premises, that a// results 

 from the same premises should be rejected. It is in vain to 

 urge, as Mr. Airy does, that the absurdity indicates the trans- 

 ition of the waves from a musical to an U7imusical condition. 

 A hiss and a roar are matters of experience and possible ; 

 they cannot therefore be symbolized by an impossibilit}'. The 

 only legitimate inference from the absurdity is, that the hj/po- 



R2 



