1883.] On the Figure of Equilibrium of a Planet, fyc. 165 



It is claimed in favour of the Laplacian hypothesis that it corre- 

 sponds to a surface density which is nearly a half of the mean 

 density of the earth, and that we know that average rock has a den- 

 sity of about 2*8. Also it is pointed out in Thomson and Tait's 

 " Natural Philosophy " that the length modulus of compressibility 

 of the surface rock is about 1/4*4 of the earth's radius, which is 

 very nearly the observed length modulus of iron. 



These conditions are not well satisfied by the present solution, for 

 the surface density is found to be '675, or 1/1*48 of the mean density 

 of the planet, whence the specific gravity at the surface is 3*7; and the 

 length modulus at the surface is equal to the planet's radius. It is to 

 be admitted that this density is large, and that the substance is also 

 highly incompressible. 



Thus in these respects the Laplacian hypothesis has the advantage. 

 It seems to me, however, that too much stress should not be laid on 

 these arguments. We know nothing of the materials of the earth, 

 excepting for a mile or two in thickness from the surface, hence it is 

 not safe to argue confidently as to the degree of compressibility of 

 the interior. There seems reason to believe that there is a deficiency 

 in density under great mountain ranges, and this would agree with 

 the hypothesis that our continents are a mere intumescence of the 

 surface layers. 



According to this view we might expect to find a rather sudden 

 change in density within a few miles of the surface. Now in any 

 theory of the earth's density such a sudden change in the thin shell 

 on the surface could not be taken into account, and the numerical 

 value for the surface density should be taken from below the intu- 

 mesoent layer if it exists. Hence it is not unreasonable to say that a 

 solution of the problem, which gives a higher surface density than 

 that of rock, lies near the truth. I do not maintain that my solution 

 is as likely as that of Laplace, but it is not to be condemned at once 

 because it does not satisfy these conditions as to the density and com- 

 pressibility of rock. 



The two cases which are given at the foot of the above table each 

 possess an interest, the first of constant compressibility, because it 

 corresponds with the case of the earth, and the second of modulus of 

 compressibility varying as the density, because this is the gaseous 

 law. 



With constant compressibility the internal ellipticity varies as the 

 *562 power of a, or nearly as the square root of the radius; with 

 gaseous compressibility it varies as the 1*562 power of a, or nearly as 

 the square root of the cube as the radius. 



A numerical comparison of the case of constant compressibility 

 with Laplace's solution for gives the following results :— 



