Wave- Motion in an Elastic Solid. 



e=B( M ).^%!; 1 



231 



(62); 



r L m w J 



where £1 == £ — 





to = t — 



. (63). 



Putting now in the equations (62) the value of B(r, t) from 

 (63), and eliminating <^ 2 (f 2 ) by (47), we find 



*«=*- 



>*-*??&{) , ^r ^i(* 2 ) 



/2 + ^3 I 2 



[« 



3qS(t,)] 





r= 



L' 



«* ^0,) ^ *z r ^,(* 2 ) 



> • (64), 



^[s^+MCtt] 



The terms of these formulas, having t x and t 2 respectively for 

 their arguments, represent two distinct systems of wave- 

 motion, the first equivoluminal, the second irrotational, 

 travelling outwards from the centre of disturbance with 

 velocities u and v. 



§ 25. I reserve for some future occasion the treatment of 

 the case in which S{t) is discontinuous, beginning with zero 

 when £ = and ending with zero when t = r. I only remark 

 at present in anticipation that ^i{t), determined by the 

 differential equation (49), though commencing with zero at 

 t = 0, does not come to zero at t = r, but subsides to zero 

 according to the logarithmic law (e~ w ) as t goes on to 

 infinity; and that therefore, as the same statement is proved 

 for ft 2 (t) by (48), neither the equivoluminal nor the irrota- 

 tional wave- motion is a limited solitary wave of duration t, 

 but on the contrary each has an infinitely long subsidential 

 rear. 



§ 26. For the general problem of the globe kept in simple 

 harmonic motion, h sin tot, parallel to OX, we may write 

 (53) for brevity as follows : — 



