1885.] third order in an isotropic elastic medium. 



307 



11. Another more general form of integral may be obtained 

 for the equation 



d 2 u 

 dz 2 



as follows. 



A first integral is 



kT-Yv 



dz) 



d 2 u 

 ~df 



da 



du 



± Jjvp 2 + k 2 dp = q + C, where p = -, 



Hence 



u = /3z + lJv& + K 2 d/3.t+ $ x (/3),1 



where z + Jv/3 2 + k 1 t + ft (J3) = Oj ' 



and <£ x (/3) is an arbitrary function of j3. Another solution is 

 obtained by changing the sign of the root and introducing a second 

 arbitrary function, thus : 



u = yz- fjvy- 4 k 2 dy . t + $ 2 (jfi), 



where z — Jvy* + k 2 t + </> 2 ' (7) = 0. 



These solutions correspond respectively to waves propagated in 

 opposite directions, and it will accordingly be sufficient to consider 

 one — say the first — of them. We must however note that they 

 cannot be superposed since the equation is not linear, or waves 

 cannot be propagated in opposite directions without affecting each 

 other. Integrating by parts — 



SJvF + k 2 dp 



= jS Jvj3 2 + k 2 + 1 [^1 log (JZ& + JW+~* 2 ) - £ JvfP -1- « 2 + G 



= /3 Jvfd? + tc 2 + function of v/3 3 + higher powers, 



^PJtf + J + xifiJ- 



Thus u = /3z + /3 Jvfi? + K* t + x(J3)t + <f>(/3) 



= cf>(/3)-/3cj>'(/3) + x (/3)t ! 

 where % (/3) shall by a proper choice of (7 be taken to vanish if we 

 put v = 0, which amounts to neglecting ( -7- ) • In this latter case 



we find </>' (/3) 4 z + id = 0, 



and u= F(z+ id). 



Thus in the case of a single term representing a wave motion 

 u = A cos n (z + /ct) = <f> (/3) - /?(/>' (/3). 



