TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 33 



It we put Z = in (137) we get a system of free annular surface- waves, in which 



n __ trl^ir- l-~ O /9 \ T / .. \ 



*yO"~ A. y ^iv A/ _ '/. | ^ j . i . ti i i Kco / 



. . . (140), 



= /^ . J 



where K is the positive root of F(f) = 0, and a] , & are the corresponding values of 

 , y6. These are of the nature of " standing " waves. 



To pass to the case of a concentrated vertical pressure Re'>' at 0,* we put in 

 accordance with (20), Z = Rfr//27r.. and integrate from to oo.f The formulae 

 (139) become 



R r^( 2 p_F-2/3) , ., w -] 

 )0 F /J i ($)<# 



.... (Hi). 

 J 



Again, the case of an internal source of the type 



+ = -*-, X=0 ..-.:.... (142), 



where r denotes distance from the point (0, 0, /), can be solved by a process similar 

 to that of Art. 5. First, superposing an equal source at (0, 0, /), distance from 

 which is denoted by r , we have 



- Mr g ihr' 



r r' ' 



and therefore, by (18), in the neighbourhood of the plane z = 0, 



(* (:-/) 



Jo(6r) 



JD a 



......... 



a 

 This makes 



This may be regarded as the kinetic analogue of BOUSSINESQ'S well-known statical problem. 



t It might appear at first sight that a simpler procedure would be possible, and that the effect of a 

 pressure concentrated at a point might be inferred by superposing lines of pressiire (through 0) uniformly 

 in all azimuths, and using the results of 7. It is easily seen, however, that such a distribution of lines 

 of pressure is equivalent to a pressure-intensity varying inversely as the distance (w) from O. This is 

 not an adequate representation of a localized pressure, -since it makes the total pressure on a circular 

 area having its centre at increase indefinitely with the radius of the circle. 



VOL. CCIIT. A. F 



