482 Mr. J. Prescotfc on the 



simple pendulum of great length. The motion actually 

 recorded is the motion which the bob of this simple pendulum 

 would have relative to the solid earth in the neighbourhood 

 of the point of observation ; for, of course, the recording- 

 apparatus moves with the solid earth to which it is attached. 

 The motion of the pendulum-bob is due to several causes of 

 which two are the attractions of the sun and moon ; tbat is, 

 the tidal actions of these bodies. If the earth yielded to 

 these attractions as freely as a liquid earth then no relative 

 motion of the pendulum and the solid earth would be 

 observed because the plumb-line would always be per- 

 pendicular to the surface. But if the solid earth were 

 perfectly rigid the plumb-line would change its direction 

 slightly with the positions of the sun and moon, and the 

 plan of the bob would trace out a certain curve on the 

 surface of the earth, which curve can be calculated from 

 the known actions of the sun and moon. It is obvious then 

 that the observed relative motion gives us the necessary 

 data for calculating the extent to which the earth yields to 

 tidal actions, and thence we can deduce a value for the 

 rigidity of the earth. It is the object of! the present paper 

 to make the necessary calculations on the yielding of elastic 

 spheres for the purpose of comparison with observations and 

 to find therefrom the average rigidity of the earth. 



So far as I know all previous calculations have been made 

 for spheres of uniform density. Now we know that the 

 density of the earth is by no means uniform throughout, and 

 it seemed very likely, beforehand that a sphere of variable 

 density might yield to a very different extent from a sphere 

 of uniform density. In this paper I have assumed laws of 

 density which agree pretty well with what we know con- 

 cerning the earth's density. Taking the density of water 

 as w I have made the mean density 5*5 w and the density at 

 the surface 2*5 u\ , 



The analysis is difficult and I have only succeeded in 

 solving the problem by assuming the sphere to be in- 

 compressible. But there would be little advantage in any 

 other assumption because we can infer from the corresponding 

 calculations for spheres of uniform density that the degree 

 of compressibility has little effect on the deformation. The 

 ellipticities calculated with the two different assumptions 

 that the earth is incompressible and that Poisson's ratio is ^, 

 differ only by about one per cent. The results are compared 

 in Love's Elasticity, Art. 183. If Poisson's ratio is put 

 equal to \ in my own result, given in Phil. Mag. Sept. 



