LAMBERT: CONSTITUTION OF THE EARTH 139 



"multi-constant" theory, there is no necessary relation between 

 the two moduli. AVithout trying to decide between the two 

 theories, it may be said that in many cases the relation indicated 

 by the "rari-constant" theory seems to hold approximately.^^ 

 If _we accept the relation as holding, then any continuous law of 

 density distribution is at the same time a law of distribution of 

 the elastic moduli. The law of density gives a relation between 

 the density (p) and the distance (r) from the center, and is like- 

 wise a relation between r and p, the pressure of the latter being 

 hydrostatic, say p — f (r) and p ^ ^p (r). The modulus of com- 

 pressibility M is defined by 



dp _ dp 

 p ~ M ' 

 By eliminating p and p, we get a relation between M and r, 

 and 3 '5 of M gives us the modulus of rigidity, m, for Legendre's 

 law of density. The values of m obtained in this way for Le- 

 gendre's law of density were shown in table i. Note that 

 the surface value of n is almost exactly what we have taken as 

 representing surface rock, and the mean value of m (averaged with 

 respect to volume) is almost exactly what was deduced from the 

 variation of latitude with Wiechert's law of density. 



These must be taken as of the nature of curious coincidences, 

 for the logic by which these values of m were found is decidedly 

 queer at first sight. If we assume hydrostatic pressure, we 

 thereby assume zero rigidity. The next step is to deduce the 

 modulus of compressibility from the law of density and the pres- 

 sure, and to take 3/5 of the modulus of compressibility to be the 

 modulus of rigidity, thus getting a rigidity quite different from 

 zero. The contradiction is less flagrant if we take into account 

 the element of time. The law of density is for pressure extending 

 over a very long time — geologic time — the compressibility is the 

 ultimate compressibility for that pressure. The tidal forces and 

 those arising in the variation have a period of a few da3^s or a 



"' That the relation holds good, or nearly so, for the matter in the earth's interior 

 is confirmed by observations on earthquake waves. See Knott, Physics of earth- 

 quake phenomena, p. 251. Oxford, 1908. Also a recent paper by him in Proc. 

 Royal Soc. Edinburgh. 39: 177. 1919. 



