171, 172.] EARNSHAW S METHOD. 211 



This is satisfied by ^ = / ( ^} , 



dt \dx) 



{/} --* 



so that a first integral of (32) is 



To obtain the complete solution of (32) we must* eliminate a 

 between the equations 



y = *c + (Cc log*) * + &amp;lt;(*)) , 



= aa:c$ + af (a) J &quot;i&quot; 



Now ?-&, 



eta p 



so that if M be the velocity of the particle x, we have 



T -c 

 whence /a = p e c . 



In the parts of the fluid not yet reached by the wave we 

 have p = p Q , u = 0. Hence we have (7=0, and therefore 





 P = P/~ C ........................... (35), 



y = ctx c log a .t + &amp;lt;/&amp;gt; (a) ............... (36), 



Q = OLX c + 2 



The function &amp;lt; is determined from the initial circumstances by 

 the equation 



*(?) . : ... 



To obtain results independent of the form of the wave let 

 us take two particles, which we distinguish by suffixes, so re 

 lated that the value of p which obtains for the first particle at 

 the time ^ is found at the second particle at the time t z . 



* Boole, Differential Equation?, c. 14. 



142 



