Wave-Motion in an Elastic Solid. 491 



r= =q-\- v (t— t) and r = q + ut, when v(t — r)>ut ; and between 

 r = q and ?' = ^ + w(£ — t), when t>r : these interesting cha- 

 racteristics of the solution of a somewhat intricate dynamical 

 problem, are secured by the particular character of the 

 originating forcive at S, which we find according to 

 § § 8, 9 to be that which will produce them. But all these 

 characteristics are lost except the first (nullity of motion 

 through all space outside the spherical surface r = q + vt), if 

 we apply an arbitrary forcive to S *, or such a forcive as to 

 produce any arbitrary deformation or motion of S. Let for 

 example S be an ideal rigid spherical lining of our cavity ; 

 and let any infinitesimal arbitrary motion be given to it. 

 We need not at present consider infinitesimal rotation of S : 

 the spherical waves which this would produce, particularly 

 simple in their character, were investigated in my Baltimore 

 Lectures, and described in recent communications to the 

 British Association and Philosophical Magazine f. Neither 

 need w r e consider curvilinear motion of the centre of S, 

 because, the motion being infinitesimal, independent super- 

 position of A'-, y-, ^-motions produces any curvilinear motion 

 whatever. 



§ 17. Take then definitively £(t), or simply S, an arbitrary 

 function of the time, to denote excursion in the direction OX ? 

 of the centre of S from its equilibrium-position. Let 'd~ 1 S 



~d~ 2 S denote 1 dtS and \ dt \ dtS. Our problem is, sup- 

 Jo Jo Jo -it, 

 posing the solid to be everywhere at rest and unstrained when 

 £ — 0, to find (|, 77, J) for every point of the solid (r>q) at all 

 subsequent time (t positive); with 



atr=0, £=£(*), V=0, 5=0 . . . (44). 



These, used in (19), give 



0= %) + m +s (irm + m-\ + \ m) ^ m \ m 



u 2 v 2 Lq[_ u v J £ 2L J 



and 



* If the space inside S is filled with solid of the same quality as 

 outside, the solution remains algebraic, if the forcive formula is algebraic, 

 though discontinuous. The displacement of S ends, not at time t=r 

 when the forcive is stopped, hut at time t = r+2q/u when the last of the 

 inward travelling wave produced by it has travelled in to the centre, 

 and out again to r=q. 



t B. A. Report, 1898, p. 783 ; Phil. Mag. Nov. 1898, p. 494. 



