MancJiester Memoirs, Vol. xlvi. (1902), No. 15. 



XV. On the Conditions which render definite the 

 Rate of Propagation of an Earth-Tremor. 



By R. F. GwvTHER, M.A. 



Received and read March tSlh, igo2. 



In a paper " On waves propagated alon^^ the plane 

 surface of an elastic solid,"* Lord Rayleigh has demon- 

 strated that, if the amplitude of the displacement 

 diminishes with increasing depth according to the simple 

 exponential law, the rate of propagation is definite and 

 unique. It was to trace, if possible, the exact mathe- 

 matical condition which enforces this unique definite rate 

 that the work of this paper was undertaken, and the 

 result is to connect it with the absence of infinities from 

 the possible forms of solution, and the consequent absence 

 of discontinuities in the stresses either within the substance 

 of the earth or upon its surface. 



Taking the surface of the earth as the plane of xy, 

 the solutions of the equations of its motion as an elastic 

 solid lying on the positive side of this plane are to be 

 sought, subject to the conditions that the stress at the 

 free plane shall vanish. 



The method which I employ, and which appears to 

 present in an advantageous form the conditions which 

 are to be satisfied when one of the co-ordinate surfaces is 

 a free surface, is to write for the components of the 

 displacement 



* Proc. Loud. Math. Soc, xyii., 1885. 



June 24th, igo2. 



