260-261] RIEMANN'S THEORY. 473 



so that both the density and the particle-velocity are propagated 

 forwards at the rate given by (9). Whether we adopt the isother- 

 mal or the adiabatic law of expansion, this velocity of propagation 

 will be found to be greater, the greater the value of p. The law 

 of progress of the wave may be illustrated by drawing a curve 

 with x as abscissa and p as ordinate, and making each point 

 of this curve move forward with the appropriate velocity, as 

 given by (9) and (11). Since those parts move faster which 

 have the greater ordinates, the curve will eventually become at 

 some point perpendicular to x. The quantities dujdoc, dp/doc are 

 then infinite ; and the preceding method fails to yield any infor- 

 mation as to the subsequent course of the motion. Cf. Art. 183. 



261. Similar results can be deduced from Earnshaw's investi- 

 gation*, which is, however, somewhat less general in that it 

 applies only to a progressive wave supposed already established. 



For simplicity we will suppose p and p to be connected by Boyle's Law 



P=c 2 P ....................................... (i). 



If we write y=x + %, so that y denotes the absolute coordinate at time t of 

 the particle whose undisturbed abscissa is x, the equation (3) of Art. 259 

 becomes 



This is satisfied by 



!=/(!) .................................. (*>. 



provided , ;/^ ...................... -(iv). 



Hence a first integral of (ii) is 



To obtain the ' general integral ' of (v) we must eliminate a between the 

 equations 



c log a) t + <j> (a), \ 



) f" 



where <p is arbitrary. Now 



dyjdx=p /p, 



* " On the Mathematical Theory of Sound," Phil. Trans., 1860. 

 t See Forsyth, Differential Equations, c. ix. 



