operative at one or more Points of an Elastic Solid. 381) 



Hertz/s solution, corresponding to the omission of the second 

 part of ;;^ in (It)), makes 



.... (22) 

 and accordingly 



cl.i- dy dz 



values which satisfy 



(V- + //0K/3,7) = (23) 



Thus (3), (4), (5) are satisfied, and the solution applies to 

 an elastic solid upon which no forces act except at the origin. 

 The onlr question remaining open is as to the character of the 

 forces which must be supposed to act at that place. This is 

 rather a delicate matter ; but it is evident at any rate that 

 the forces are not of the simple character contemplated in the 

 preceding investigation. It would appear that they must be 

 double or multiple, and have components parallel to x and y 

 as well as z. By a double force is meant the limit of a 

 couple of given moment when the components increase and 

 their mutual distance decreases, analogous to the double 

 source of acoustics. 



I now propose to calculate the work done by the force Z^ 

 at the origin as it generates the waves represented by (18), 

 (19). For this purpose we require the part of y in the 

 neighbourhood of the origin which is in quadrature with the 

 force, i. e. is proportional to sin pt. From (19) we get 



XW^inptri, ^2 3c^ 1 \ . , /I 3^^^\ , -1 



. . . (24) 

 the last term (in cos pt) not contributing. Expanding sin kr, 

 cos kr and retaining the tei-ms of order h\ we get for the 

 square bracket 



r^^r+^JO). 



Thus 



(21) = p3Asiny>^, 



when r is small, so that the part proportional to sin pt is in 



