Mr. R. H. M. Bosanqiiot on the Tlieory of Sound. 31 

 Qvevy surface S in the same time dt ; .*. SJr= const, di, and 



sv=s„v„. 



The energy of the flow under these circumstances lying be- 

 tween surfaces S^ and S, is 



When r becomes great in comparison with r^, the last term 

 ceases to influence the expression. Consequently the energy 

 of the whole motion lying beyond such a value of r is negligi- 

 ble in comparison with that which lies withni it ; or if we 

 make r indefinitely great, the energy has the finite value 

 (S„ = 4,rrJ) 



None of this passes off to infinity ; consequently the mainte- 

 nance of the motion when once started requires no additional 

 expenditure of energy (of course friction is neglected). Hence 

 we have the curious theorem^ that if an infinite mass of fric- 

 tionless fluid be set in symmetrical divergence in the manner 

 indicated, as soon as steady motion has set in no further 

 pressure is required from the supply-channel, but if the fluid 

 is brought to the surface of the small sphere from which diver- 

 gence takes place it will be drawn out, and the motion conti- 

 nued, by means of the energy already present. 



I will now endeavour to analyze more closely the course 

 taken by the flux of energy in this problem, and to illustrate 

 the mechanical action by which this last result is produced. 



We may suppose the energy to be divided into two streams, 

 both accompanied with outward velocity of displacement. 

 The one is divergent ; it has a greatest pressure inside, and 

 a less pressure towards the outside. The other is convergent; 

 and it proceeds inwards with continually increasing rarefactions. 



If we combine these two streams in the same way as the 

 two opposite streams of sound which make up a stationary 

 vrave (see Prop. 1. of note 4), the velocities are both equal to 



— at any point, and the condensations are 



and 



V 



4- -r- for the diver o^ent wave, 



Y 



77- for the convergent wave. 



