TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 11 



where / = </{a? + (y +/) 2 }. It is evident, without calculation, that the condition 

 of zero tangential stress at the plane y = is already satisfied ; the normal stress, 

 however, does not vanish. It appears from (13) that in the neighbourhood of the 

 plane ?/ = "the preceding value of <f> is equivalent to 



_ i p er<*-f> (ft* dg -If" 



IT J TT J_ 



TT _ a 



2 f" cosh 0.1 



f , tr 



e e 



Substituting in (29) we find that this makes 



a 



Comparing with (46), we see that the desired condition of zero stress on the 

 boundary will be fulfilled, provided we superpose on (57) the solution obtained from 

 (30) and (48) by putting 



TT a 



and afterwards integrating with respect to f from -co to co . The surface- 

 displacements corresponding to this auxiliary solution are obtained from (51), and if 

 we incorporate the part of U Q due to (58), Ave find, after a slight reduction, 



...... (GO). 



These calculations might be greatly extended. For example, it would be easy, 

 with the help of Art. 4, to work out the case where a vertical or a horizontal periodic 

 force acts on an internal line parallel to z. And, by means of the reciprocal theorem 

 already adverted to, we could infer the horizontal or vertical displacement at an 

 internal point due to a given localized surface force. 



6. It remains to interpret, as far as possible, the definite integrals which occur in 

 the expressions we have obtained. 



It is to be remarked, in the first place, that the integrals, as they stand, are to a 

 certain extent indeterminate, owing to the vanishing of the function F (f ) for certain 

 real values of f. It is otherwise evident d priori that on a particular solution of any 

 of our problems we can superpose a system of free surface waves having the wave- 

 length proper to the imposed period '2-ir/p. The theory of such waves has been given 



C 2 



