406 Mr. Challis on the general Equations 



density will return to the same values after equal intervals of 

 time; and it would seem from the equation above, that the 



interval for the velocity w is — ; — ; whereas it should be — since 



a + w a 



the propagation is uniform. But it is to be observed that the 

 equations of the motion are also satisfied by 



w = a hvp. I02:. p = m sin - ( at), 



from which it may be inferred that at a given point, the instants 

 at which the same velocity recurs are separated by the constant 



interval of time — . The two integrals compared together, shew 



that in the latter X may either be constant, or may vary inversely 

 as a + w. that it is constant, when the variation of velocity and 

 density at a given point is to be determined, and varies inversely 

 as a + w, when the variation of velocity and density from one point 

 to another, is to be determined at a given instant. A similar 

 observation may be made with respect to the other integral. 



I will here observe that the form of the arbitrary function 

 found above, indicates the kind of vibration which Newton 

 selected in his theory of sound, without giving any reason for 

 his selection. He might have inferred from his law of the 

 vibrations, that two propagations could obtain simultaneously 

 in opposite directions, and appealed to experience for a confir- 

 mation of the truth of the inference. In the analytic theory the 

 process of reasoning is in the reverse order ; the possibility of 

 the simultaneous propagations is first proved, and thence the 

 law of the vibrations is deduced. This requires the solution of a 

 functional equation, according to the principles we have exhibited. 



7. Now let a con.stant force g act on the fluid. The equa- 

 tions (r), Art. 5. for this instance become, 



s — at - fuidt = c. 



