Prof. Challis on the Hydrodynamical Theory of Vibrations. 301 



by Mr. Airy (Phil. Mag. for June 1849, p. 401) is, that a given 

 wave continually changes type till it is eventually broken up, so 

 that musical sounds are by mere propagation through space con- 

 verted into unmusical noise. The interpretation of Professor 

 Stokes (Phil. Mag. for November 1848, p. 349) is not essentially 

 different from this. It is not asserted that this interpretation is 

 in accordance with observed facts. Moreover, there lies against 

 it the obvious objection that the very same equations which indi- 

 cate that musical sounds become unmusical, equally indicate 

 that the latter again become musical, and so on alternately. 

 This is so great an absurdity, that, as there is no reason to 

 question the accuracy either of the principles on which the above 

 equations rest, or of the mathematical process by which they were 

 obtained, the conclusion is inevitable that those principles are 

 insufficient. I have supplied the defect by the discovery of a 

 third fundamental equation, by means of which it has been 

 proved, in this communication, that, in conformity with experi- 

 ence, undulations are propagated through space uniformly and 

 without undergoing change. 



48. Having now gone through a revision of my hydrodyna- 

 mical theorems, so far, at least, as they relate to the approxima- 

 tion of the first order, I have only to add a few general remarks. 

 All the foregoing investigations have had reference to the solu- 

 tion of a problem which may be thus enunciated : — To deter- 

 mine the motions of a continuous mass of unconnected particles 

 subject to given conditions. The investigations have had nothing 

 in them speculative or conventional. The premises are matters 

 of fact on which all mathematicians are agreed ; and the required 

 results are to be obtained by reasoning from right principles 

 according to logical rules. If two mathematicians differ in any 

 conclusion, it is because there has been erroneous reasoning on 

 one side, or on both. In a question to be answered by pure 

 reasoning, unanimity ought to be attainable. The problem is 

 different from that of determining the motion of a continuous 

 mass of rigidly connected particles subject to given conditions, 

 which can always be effected by means of ordinary differential 

 equations, being, in fact, virtually the determination of the 

 motion of a single material particle acted upon by given forces. 

 The other problem demands the formation of differential equa- 

 tions of a more comprehensive kind, containing at least three 

 variables; and it is in the application of such equations that the 

 difficulty of solving it lies. Newton discovered the principles 

 and calculations proper for determining the motion of a single 

 particle acted upon by given forces; the discovery of the prin- 

 ciples and calculations required for determining the simultaneous 

 movements of the unconnected particles of a continuous mass 



