Gravitational Problem. 93 



agreeing with those of the surface. There would accordingly 

 be no periodic term in the reaction between the body in 

 question and the ground upon which it rested ; the weight 

 of the body would appear perfectly constant, and, generally, 

 no observations concerned wholly with gravitating bodies at 

 the earth's surface * would give any indication of periodic 

 variations iu the gravitation u constant." 



6. Now further add to the system of forces acting on the 

 earth a superficial periodic pressure — P ; thus the sole 

 periodic influence at work is that due to the term GA^ in 

 the gravitation "constant," while the only effects observable 

 at the earth's surface are such as arise from the pressure —P. 

 The result of such a periodic pressure is a radial movement 

 of the surface of period 2ir/p^ and of an amplitude which, 

 under certain simplifying assumptions, can be readily ex- 

 pressed in terms of the elastic constants of the earth's 

 substance. 



7. Homogeneousness has already been assumed ; let us 

 now further postulate some small degree of viscosity, suffi- 

 cient to ensure that compressional waves of short period (one 

 second or less) propagated inward from the surface, would be 

 sensibly extinguished before reaching the centre, though not 

 sufficient to produce any perceptible damping effect in the 

 course of a few wave-lengths. We have thus only to consider 

 the waves travelling inward from the surface under the in- 

 fluence of the periodic pressure — P ; and since, in the cases 

 to which this inquiry is restricted, the wave-length of the 

 disturbance is very small compared with the earth's radius, 

 the curvature of the surface may be disregarded, and the 

 problem of the motion near the surface reduces to one of 

 plane waves. 



8. Let a — r, the depth of any point below the surface, be 

 denoted by h ; the waves due to the pressure — P and propa- 

 gated in the direction of ^-increasing may be represented by 



Sh = LV'M<-*< V >, (6) 



where the constant C has to be found and is not necessarily 

 real ; while the velocity V is given by 



V 1 « (*+*0/f> ; (7) 



Here k, as already denned, is the bulk-modulus of elasticity 

 of the earth's substance, while fi is its rigidity, and p its 



* The attractions of suck bodies for one another are here excluded 

 from consideration. 



