THE PRESSURE WITHIN THE EARTH 75 



The decrease in the size of the earth that would be brought 

 about by the increase of internal pressure discussed above, may 

 be computed, if it be assumed that the high density in the inte- 

 rior of the earth is due alone to the compressibility of matter 

 under the enormous pressures there present. Laplace, as a 

 matter of fact, began by assigning a b.w of compressibility 

 and thence deducing the law of density; however, his law of 

 density might be close to the truth and yet be not entirely con- 

 trolled by the pressure. The equation 



r a a . 2.46015 a 



P = 4-3 6 5 - sm 



a a 



leads to the following relation connecting pressure and density. 1 



dp = k d (p 2 ) 



or, as stated by Laplace: The variatio?i in pressure in the interior 

 of the earth is proportional to the variation in the square of the density. 



The law of compressibility of gases — "Boyle's Law" — and 

 the law of elasticity for small compressions in solids — " Hooke's 

 Law " — state that the variation in density is directly proportional 

 to the variation in pressure. Thus Laplace's law assumes a com- 

 pressibility which is less than that given by either of the laws 

 just named, an assumption which is, in itself, very reasonable. 



From the above equation we derive 

 k p- =p -f- c 

 and determining the constants on the supposition that the sur- 

 face density is 2.75, and the central density 10.74, we conclude 

 that 



(.02575) p 2 =p-\- . 1947 



in which p must be given in terms of a million atmospheres as 

 unit. To determine the change in density due to a small change 

 in pressure we may write, 



dp d p 



- 2 



p + c p 



\T+~c) 



x See O'Brien, Math. Tracts, p. 39; Pratt, p. 113; Thompson and Tait,TI, p. 403. 



= 2 T P 



