1881.] Stresses caused in the Interior of the Earth, fyc. 433 



The existence of dry land proves that the earth's surface is not a 

 figure of equilibrium appropriate for the diurnal rotation. Hence the 

 interior of the earth must be in a state of stress, and as the land does 

 not sink in, nor the sea-bed rise up, the materials of which the earth 

 is made must be strong enough to bear this stress. 



We are thus led to inquire how the stresses are distributed in the 

 earth's mass, and what are magnitudes of the stresses. 



In this paper I have solved a problem of the kind indicated for the 

 case of a homogeneous incompressible elastic sphere, and have applied 

 the results to the case of the earth. 



If the earth be formed of a crust with a semi-fluid interior, the 

 stresses in that crust must be greater than if the whole mass be solid, 

 very far greater if the crust be thin. As regards the condition of in- 

 compressibility attributed to the materials of the earth, it is proved in 

 this paper that the compressibility of the solid would make no sensible 

 difference in the results ; except, indeed, in the case where the defor- 

 mation of the sphere is of the second spherical harmonic class, when 

 large compressibility would considerably modify the results. . 



The strength of an elastic solid is here estimated by the difference 

 between the greatest and least principal stresses, when it is on the 

 point of breaking, or, according to the phraseology adopted, by the 

 breaking stress-difference. The most familiar examples of breaking 

 stress-difference are when a wire or rod is stretched or crushed until 

 it breaks ; then the breaking load divided by the area of the section 

 of the wire or rod is the measure of the strength of the material. 

 Stress- difference is thus to be measured by tons per square inch. 



Tables of breaking stress-differences for various materials are given 

 in the paper. 



The problem is only solved for the class of inequalities called zonal 

 harmonics ; these consist of a number of waves running round the 

 globe in parallels of latitude. The number of waves is determined by 

 the order of the harmonic. In the application to the earth the equator 

 here referred to may be any great circle, and is not necessarily the 

 terrestrial equator. The second harmonic has only a single wave, and 

 consists of an elevation at an equator and depression at the pole ; this 

 constitutes ellipticity of the spheroid. An harmonic of a high order 

 may be described as a series of mountain chains, with intervening 

 valleys, running round the globe in parallels of latitude, estimated 

 with reference to the chosen equator. 



The case of the second harmonic is considered in detail, and it is 

 shown that the stress-difference rises to a maximum at the centre of 

 the globe, and is constant all over the surface. The central stress- 

 difference is eight times as great as the superficial. 



On evaluating the stress-difference arising from given ellipticity in 

 a rotating spheroid of the size and density of the earth, it appears that 



