On the Production of Ware- Motion in an Elastic Solid. 389 



where e denotes what I call the " stiffness " of the spring- 

 system. We may now conveniently write (72') short as 

 follows : — 



P = h(asmctt + bcosut) . . . (72")- 



Explicit expressions for a, b are given in (91) below, with Q 

 taken to denote %"np<f. 



§ 34. ¥ove, (83) and (72") give 



< , = /ij(lH JsinwH — coswn . . . (84); 



and with this in (82) we find 



»*«* ( 1 A J sin at -\ — cos tat 



= (a +7&) - J sin (at+ b — ycal 1 H J sin cat f 



which requires that 



(1 + — ] mm 1 = a-\ ya>, and — m&> 2 = h—(l-\ 1 7&> ; 



by which, solved for two unknown quantities, y&> and m» 2 , we 

 find 



r°>= {a + cf + b» < 85 >» 



and 



, c[a(a+c) + b 2 ] 



(a + cf + b* K 1 



If we suppose <o and c known, these equations, with (91), 

 for a and b, tell what w/Q must be in order that the force 

 applied to maintain the periodic motion of the sheath shall 

 vary in simple proportion to the velocity; and give y, the 

 magnitude of this force per unit velocity. 



§ 35. If we denote by E the maximum kinetic energy of m, 

 we find immediately from (84), 



E=pw[(i+^y+(^yj . . . («7). 



And by (73') we have, for the work per period done on the 

 sheath by P, 



rw = ^rh t o)b (88). 



