414 JOURNAL OF THE WASHINGTON ACADEMY OF SCIENCES VOL. 13, No. 19 
mean density of the Earth to the correct value. A number of empiri- 
cal laws have been suggested for the increase of density with depth. 
The best known of these is the one proposed by Laplace. According 
to this the density at any distance r from the Earth’s center is given 
by the equation 
Pp = — —— | (1) 
in which pp is the density at the center and q is a constant of which 
the value is fixed by the known total mass of the Earth. Another 
well-known relation is that of Roche: 
p = po (1 — kr’) (2) 
in which k is a constant which also can be determined from the total 
mass or the mean density of the Earth. Either of these formulas, 
with the usually assumed surface density 2.7, indicates a density at 
the center somewhat above 10. 
The increased density at the center obviously may be due either to 
the presence of heavier material, presumably iron or nickel-iron, or to 
a diminution of volume by the tremendous pressure existing at great 
depths—or both factors may enter. It has often been assumed that 
the increase of density with depth is merely the result of the com- 
pressibility of the homogeneous material, and that the Laplace law, 
for example, could be used to calculate the compressibility of the 
Earth at the surface and in the interior. There is no a priori reason 
why this could not be so, but clearly other lines of evidence must be 
examined before an answer to this question can be secured. 
Moment of inertia of the Earth. It is obvious that for a given mass 
(or for a given mean density) the moment of inertia depends on the 
distribution of density,’ e.g. if there is heavy material at the center 
and light material at the surface the moment of inertia would be 
considerably less than if the central density were smaller than that 
of the surface. The moment of inertia itself is not sufficient to fix 
’ The moment of inertia of a sphere with its mass symmetrically distributed about 
the center is 
Oo ee 
in which p is the density at distance r from the center. For a homogeneous sphere this 
becomes 
Sr 
C= apr=04 Mr 
15 
M being the total mass. 
