170, 171.] RIEMANN S METHOD. 200 



Multiplying (28) by A// &amp;gt; adding to (27), and writing for 



shortness 



, . &amp;lt;/? / fdp\ dr 



we obtain -31 = \ u + A / ^r I T &amp;gt; 



dt V V &amp;lt;W &amp;lt;fo 



rfs 



di ~~ 



whence rfr = J {& - ( + J ) cfi ............... (30), 



If the values of r and s can be found, those of u and /&amp;gt; follow at 

 once from (29). Now (30) and (31) shew that r is constant for 

 a geometrical point moving with velocity 



dx _ /dp 



_ 

 t~ 



whilst 5 is constant for a point moving with velocity 

 dx _ /dp 



= 



Hence any given value of r travels forward with the .velocity 

 f + u t and any given value of s backwards with the velocity 



These results enable -us, if not to calculate, still to understand 

 the character of the motion in any given case. Thus let us 

 suppose that the initial disturbance is confined to the space 

 between the values a and b of x; so that we have initially for 



x &amp;lt; a, ? = r 1? s = s t , 

 and for x &amp;gt; 6, r r z , s = 5 2 , 



I, 14 



