232 Mr EARNSHAW, ON FLUID MOTION. 



Hence, at points of mean density the particles are stationary; at points 

 of condensation the particles are moving forwards ; and at points of 

 rarefaction they are moving backwards. 



34. The property which was proved in Art. 23, respecting plane- 

 waves, may be extended to curvilinear waves; and we may shew gene- 

 rally, 



That no wave-surface, terminated ahrupthj hy sharp edges, can be 

 transmitted through a medium unless its edges rest upon the boundaries 

 of the medium. 



For, the only expression for a curvilinear wave which the equation 

 of continuity furnishes is 



<p = ^.F{f{x-a)+g{y-li)-\-h{,z-^)-at,f,g, h]; 



which, as before observed, teaches that we may suppose the wave com- 

 posed of plane-waves, which we may suppose transmitted through the 

 medium, and then we shall have the true wave-surface by taking that 

 to which they are all tangents. 



Now, suppose a wave-surface terminated abruptly to be by some 

 means or other excited in a medium. Upon referring to the above 

 expression for cf>, we should find that the tangent-planes at the edges 

 of the wave-surface, or rather the component waves, represented by 

 these tangent-planes, and expressed by terms of the form 



F.{f{x-a)+g'{y-(i)+h'{%-y)-at,f, g'] h'\, 



stretch out indefinitely beyond the boundary of the wave-surface into 

 the medium : and when these components are transmitted and afterwards 

 compounded into one wave, the portions of these waves which (as it 

 were) hung over the proper wave-surface must remain. Hence it 

 appears. 



First, That a wave-surface terminated abruptly by sharp edges cannot 

 be excited in a medium : and 



