12 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



by Lord RAYLEIGH,* and is moreover necessarily contained implicitly in our 

 analysis. 



Thus, if we put Y = in (47), we find that the conditions of zero surface-stress are 

 satisfied, provided 



A : B = 2/r k~ : 2iKOi l = 2i.K/3 1 : 2* c k z . . . . (61), 



where K is a root of F (f ) = 0, and 1( /8 ls denote the corresponding values of a, /3, 

 Now, in the notation of (49) and (50), 



Equating this to zero, we have a cubic in f 8 /^, and since k~ > /r, it is plain that 

 there is a real root between 1 and oo . It may also be shown without much difficulty 

 that the remaining roots, when real, lie between and h~/k z . The former root makes 

 a, ,5 real and positive, and therefore cannot make f() = 0. The latter roots make 

 a, /B positive imaginaries, and therefore cannot make F (f) = 0. This latter 

 equation has accordingly only two real roots i K , where /c > I;. 

 Thus, in the case of incompressibility (X = oo , h= 0) it is found that 



K/k= 1-04678 



and that the remaining roots of (62) are complex, t On POISSON'S hypothesis as to 

 the relation between the elastic constants (X = JM, 7r = ^P), the roots of (62) are all 

 real, viz., they are 



eiV = \, U3-V/3), i(3 + v/3), 

 so that 



K /k = i v/(3 + v/3) = 1-087664 . . . ; 



this will usually be taken as the standard case for purposes of numerical illustration. 

 In analogy with (28), it will be convenient to write 



=pc ........... (63), 



where c denotes the wave-slowness of the Rayleigh waves. The corresponding 

 wave-velocity is 



k 7 _-, k /u, 



c l =.b l =. A / C . 



K K V p 



According as we suppose X = QO , or X = /x, this is '9553 times, or '9194 times, the 

 velocity of propagation of plane transverse waves in an unlimited solid. 



The further properties of free Rayleigh waves are contained in the formulae (61) 



* 'Proc. Lond. Math. Soc.,' vol. 17 (1885); 'Scientific Papers,' vol. 2, p. 441. 



t Cf. RAYLEIGH (loc. tit.), where it is also shown (virtually) that they are roots of / (), not of F (), if 

 a, /3 be chosen so as to have their real parts positive. 



