246 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1923 



and by Knott. 12 The velocity of both kinds of waves increases rap- 

 idly at first, and then steadily and almost linearly until a depth of 

 1,600 km. is reached, after which the velocity, although nearly con- 

 stant, shows a tendency to fall off, especially at about 3,000 km. As 

 will be shown below, when the density at various depths is known, 

 these curves can be converted into compressibility-depth and rigidity- 

 depth curves. 



DENSITY CHANGE DUE TO COMPRESSION 



We shall next use the above results to determine to what extent 

 the higher density of the interior of the earth may be due to com- 

 pression alone. The decrease in volume caused by pressure at great 

 depths can not be calculated from the measured compressibility of 

 rocks, even if the pressure were known, because the compressibility 

 decreases with the pressure, which at a depth of only a few hundred 

 kilometers is far beyond the range of laboratory measurement. But 

 fortunately, the velocity of transmission of earthquake waves yields 

 information as to the variation of compressibility (1/K) with depth. 

 The values of K/p at various depths have been calculated from the 

 earthquake velocities (see footnote on p. 245), and the results shown 

 in column 4 of Table 1. Now, it is reasonable to suppose that from 

 this information concerning compressibility it would be possible to 

 determine the aggregate diminution in volume at a given depth 

 on the supposition of a homogeneous earth whose central density 

 is made high by compression and not by a change of composition. 

 An equation connecting these quantities has been derived, 13 and used 



"Proc. Roy. Soc. Edinburgh, 39, 167. 1918. 

 13 The equation is derived as follows : 



In general, 



dp ri.66X10-«TOp 



dT 0P= rT— 



where g is the acceleration of gravity and p is the pressure at distance r from the center; and m, the 

 mass of tho sphere of radius r, is obtained from the relation 



i=4*r Ipr'dr 



Now the first equation may be written -?■ • -r— — : - } — — 



but, on the assumption of homogeneity, p -^"K, by definition. 



Therefore, by division 



dlnp 6.66yi0-«mp 

 dr " r'K 



or, r being the mean radius of the earth and pr the surface density, 



r 



' n ~— — I — — -j^ — - dr, which is the desired expression. B 



r 

 The density-depth relation is obtained from this equation by approximation and re- 

 peated graphical integration. First, the density at various levels is assumed (consistent, 

 of course, with the known average density of the earth). The quantity, p r 2 , is then 

 plotted against r, and m found by graphical integration of equation A. Next, the quan- 

 tity, m p/f*K, is plotted against r, and p as a function of r determined according to equa- 

 tion B by another graphical integration. This first approximation for p is used to 

 calculate a new curve for m, which in turn yields a second approximation for p. It 

 turns out that the convergence is very rapid, so that with almost any initially assumed 

 values of the density three successive integrations are sufficient. 



