1890.] of the Rigidity of the Earth. 73 



that in replacing the earth by an elastic solid mass of gravitating 

 matter we may perhaps be enabled to estimate better the amount 

 of the tidal distortion if we assume this relation to hold. I have 

 thei-efore solved the following problem — A mass of solid matter, 

 homogeneous in the natural state, free from all applied forces, and 

 filling a spherical surface, being given, such solid is strained by its 

 own gravitation according to the Newtonian law, and by the 

 application of external disturbing forces having a potential ex- 

 pressible in spherical harmonic series. Supposing the deformed 

 free surface expressed as a harmonic spheroid, it is required to find 

 the amount of the harmonic inequality. — The notation of Thomson 

 and Tait's Natural Philosophy is used and the solutions of the 

 parts of this problem there considered are adopted. The problem 

 has not been previously solved with the generality here considered. 

 Now the general equations of equilibrium of the solid are 

 three, of the type 



in 



doc 



+ /,V 2 a + pX = (a), 



and it is well known that the solution consists of two parts, (1) any 

 set of particular integrals of these equations, and (2) such comple- 

 mentary functions satisfying a system identical with (a) when the 

 terms depending on the bodily forces such as X are left out as 

 with the set of particular integrals (1) will satisfy the condition 

 that the external surface remains free from stress after the defor- 

 mation. The complementary functions for our problem are given 

 in Thomson and Tait, Art. 736, and the method here used for 

 obtaining the particular integrals is the same as that of their 

 Art. 834, but the surface conditions cannot be immediately written 

 down by their method. This happens for two reasons — firstly, 

 because the stress arising from the attraction of the mass is so 

 great compared with the other stresses, that its amount has to be 

 estimated at the external deformed surface and not at the mean 

 spherical surface as the others may be, and secondly, because part 

 of the system of bodily forces consists in the attraction of the har- 

 monic inequalities whose expression involves the complementary 

 functions. It is however easy to surmount the latter difficulty by 

 forming an equation giving the potential of the harmonic inequali- 

 ties in terms of the disturbing potential and quantities that occur 

 in the expression of the complementary functions. The method of 

 estimating the surface tractions that must be regarded as applied 

 to the mean sphere in consequence of the attraction of the mass, I 

 have considered in a previous paper (Proc. Lond. Math. Soc, Xix. 

 pp. 185 sq.) in the case of vibrations, and a like method applies 

 here. The total surface traction arising from complementary 

 functions and particular integrals is thus found and resolved into 



