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XVII. Additional Analytical Considerations respecting the Ve- 

 locity of Sound. By the Rev. J. Challis, M.A., F.R.S., 

 F.R.A.S.) Plumian Professor of Astronomy and Experi- 

 mental Philosophy in the University of Cambridge*. 



■ PROPOSE in this communication to resume the inquiry 

 *■ respecting the mathematical determination of the velocity 

 of sound, at the point where it was left (Phil. Mag. Supple- 

 mentary number, p. 498) in my reply to the remarks made by 

 the Astronomer Royal on my communication in the Number 

 of the Philosophical Magazine for April. 



After proving (Phil. Mag. for April, p. 277) that the expres- 

 sion udx -f- vdy ■+ isodz must be made integrable in a general 

 manner, it was shown (Supplement, pp. 496 and 497) that on 

 satisfying this condition by supposing the velocity to be a 

 function of the distance either from a plane or from a centre, 

 contradictory results were arrived at, which necessarily led to 

 the inference that neither plane- waves nor spherical waves are 

 physically possible. A third supposition, according to which 

 the above expression is the differential of the product of a 

 function (<p) of z and t, and a function (/) of xnndy, involved 

 no contradiction, the reasoning being confined to the first 

 order of approximation. To complete the consideration of 

 this third case, which for the sake of distinction may be called 

 that of non-divergent waves, it will be necessary to inquire 

 whether any contradiction is met with when the reasoning is 

 conducted by exact equations, which I now proceed to do. 



The reasoning by which it was concluded that udx + vdy 

 + ivdz must be made integrable in a general manner, went on 

 the supposition of small vibratory motions, and quantities of 

 a higher order than the first were neglected. I have elsewhere 

 urged (Camb. Phil. Trans., vol. viii. part 1, p. 35 ; and Phil. 

 Mag. S. 3. vol. xxi. p. 426) that in this process of reasoning 

 the equations 



du _ dv du _ dw dv _ dw 



dx dy* dz~ da?' dz dy 



are approximate, whereas the inference of integrability requires 

 that they should be identical equations. With reference to 

 this point, Mr. Stokes, in his Report on Hydrodynamics (Re- 

 port of the British Association for 1846, p. 2), has correctly 

 remarked, that the reasoning proves the quantity in question 

 to be approximately an exact differential, which implies that 

 there exists an expression u 1 dx-\-v 1 dy-{-'w 1 dz which is accu- 

 rately an exact differential, u v v v w ] differing from u, v, w 

 * Communicated by the Author. 



