Prof. Challis on the Nature of Aerial Vibrations. 463 



cally possible ; and that as these become plane-waves at an 

 infinite distance from the centre, the latter are also physically 

 possible. I have already met this argument in the communi- 

 cation above referred to ; but as the reasoning is given briefly, 

 and may possibly have been overlooked, I will repeat it here. 

 I take the results of the hypothesis of spherical waves as they 

 are given in Poisson's Traite" de M.e'canique (vol. ii. p. 706, 

 2nd edit.), and as they are commonly admitted. The pressure 

 p being « 2 (1 + <r), the following expression is obtained for the 

 condensation <r at the distance r from the centre at the time t, 



ar 



and it is stated that there is no condensation wherever r is 

 greater than at + e, and less than at—s, 2e being the breadth 

 of the sonorous undulation. Hence, supposing 2s very small 

 compared to r, and putting for r outside the function, its value 

 at corresponding to the middle of the wave, the quantity of 

 matter existing at any time in the wave beyond what would 

 occupy the same space in the quiescent state of the fluid, is 

 very nearly 



aH ' 



the integral being taken from r—at — e to r=at + s. Calling 

 A the constant value of this integral, the expression for the 

 quantity of matter becomes 4nr At. Hence the matter increases 

 in quantity with the time ! Now the very equations from which 

 this result is derived are founded on the supposition that the 

 quantity of matter is constant. There is consequently no dif- 

 ficulty here which any physical considerations can explain, 

 but strictly a reductio ad absurdum, which necessitates the im- 

 portant conclusion that the hypothesis of spherical waves is 

 inadmissible. The physical impossibility of plane-waves was 

 proved by the same kind of reasoning ; and any attempt to 

 reconcile the contradiction in either case is simply illogical. 

 As neither the hypothesis of plane-waves nor that of spherical 

 waves is admissible, the theoretical value of the velocity of 

 sound which rests on those hypotheses necessarily fails of sup- 

 port. I return now to the consideration of non-divergent 

 waves, or, as they may also be called, ray-vibrations. 



The general equation which gives the density p in any in- 

 stance of fluid motion, the velocity V being known, is 



/dV V 2 



Tt ds+ ~=F(a 1 V 1 /). 



