8 GWYTHER, Rate of Propagation of an Earth-Tremor. 



Since we have ai^a.^, and bi = bi, we may make a 

 considerable simplification in the components of the 

 displacement in each kind of wave. In the case of the a's, 

 we shall have 



t; = ^^{27^-(i +tanh2y)/}. 



2i^-(l+COth2y)/} . . . . (ll), 



where F and / only differ by containing /3 and 7 res- 

 pectively, and where the form of the function is 



[F{U+iV) + F{U-iV)\ 

 Similarly, in the case of the b's, we shall have 



4{= 



u = B^ < coth/3cothy(T + tanhV)^' - 2/' j , 

 7^ = B^{ coth/3coth7(i + tanhV)^' - 2/' [ , 



w = £^l coth/3cothy(i +tanh2y)-^'-2COth2y/' I (12), 



where, again, F' and/' only differ by containing j3 and y, 

 but where the form of each function is 



[F'(l7+tV)-F'{U-iF)]/t. 

 The condition that the waves should be surface waves, 

 so that the amplitude diminishes rapidly with increasing 

 depth and tending asymptotically to zero, can be satisfied 

 by a very varied selection of functions, both algebraic and 

 transcendental. We may construct functions which will 

 give for the displacement apparently infinite values 

 either at single detached poles or at a periodic series of 

 poles, either simple or of higher order at the plane of xy. 

 If, however, we are to exclude all such infinities, 

 we are left only with the single case of diminution of 

 amplitude according to the simple exponential law 

 investigated by Lord Rayleigh. 



