266 The Rev. Samuel Haughton on the 



There remains, however, the third integral on the left-hand side of (14), 

 which cannot be known without assuming a law of density and also of ellipti- 

 city for the solid portion of the Earth. 



We are thus led to the conclusion, that it is necessary to assume titree hy- 

 potheses with respect to the internal structure of the Earth, before we can be in 

 a position to assert how far it is solid and how far fluid. The three necessary 

 hypotheses are : — 1st. The law of density of the fluid parts. 2nd. Tlie law of 

 density of the solid parts. 3rd. The law of ellipticity of the solid parts. 



If we suppose tliat these are given, then equations (14), (15) will become, 



i^(a,a,, £, ei, m) = 0; 



(17) 

 /(a, a,, e, ei, m) = 0; 



in which F,f denote known functicms. In these equations a, e, m are known, 

 and ai, ej, are determined by the equations themselves. 



If we suppose that the fluid parts of the earth are bounded on both surfaces 

 by solids, we should then have three equations, analogous to (14) and (15), be- 

 longing to the two surfaces of the fluid, and to the external surface respectively. 

 From these, assuming the law of density of the fluid, and of density and ellipti- 

 city of the solid parts, we should obtain 



^ (a, ai,a,, c, ei,e.,, m) = ; 



X (a,ai,a,,e,ei,e,,m) = 0; (18) 



•^•(a, ai.a.,, e, ei,e2, m) -0 ; 



3,2 , £2 being the radius and ellipticity of the second surface of the fluid. In 

 equations (18), as before, a, e, m are known; but the number of unknown 

 quantities is greater than the number of equations, the unknown quantities 

 being four, viz., a^, a.^, cj , e,, while there are only three equations. The pro- 

 blem is therefore not so definite as the last, and requires an additional hypo- 

 thesis. 



Confining our attention to the simplest case (17), we see that before a 

 single step can be made towards using equations (14) and (15), we must as- 

 sume three laws, respecting facts of which we have no certain knowledge, 

 and probably never shall. The subject would thus appear to be excluded 



