Wace- Motion in an Elastic Solid. 



487 



Supposing v > u, we see that these two durations overlap 

 by an interval equal to 



r — q r—q \ 



V u 



if 



r — q<T 



1 [u " v) 



(32) 



On the other hand, at every point of space outside the radius 

 q + T/(llu — l/v) the wave of the greater propagational 

 velocity passes away outwards before the wave of the smaller 

 velocity reaches it, and the transit-time of each wave across 

 it is t. The solid is rigorously undisplaced and at rest 

 throughout all the spaces outside the more rapid wave, 

 between the two waves, and inside the less rapid of the two. 



§ 12. The expressions (25) and (26) for the components of 

 the surface-forcive on the boundary of the hollow required to 



produce the supposed motion, involve $y and <%• Hence we 

 should have infinite values for £ = or t = r, unless F x and F 2 

 vanish for 2 = and t = r, when r — q. Subject to this con- 

 dition the simplest possible expression for each arbitrary 

 function to represent the two solitary waves of § 11, is of the 

 form 



r =(i-x 2 ) 3 



where %= — — 1 



(33). 



Hence, by successive differentiations, with reference to t, 



12 



P=-^x(i~x* 



1 



24- 

 ^ = ^(-l + 6^-5 x 1 ) 



(34). 



^ = ^(12x-20 X s 



J 



The annexed diagram of four curves represents these four 

 functions (33) and (34). 



§ 13. Take now definitively 



Fi (' - !0 =Ci!?3(1 ~* i2)l 



(35), 



2 L2 



