THE LAPLACIAN LAW OF DENSITY. 87 



III. THE LAPLACIAN LAW OF DENSITY. 



In treating the rotations of such bodies as the earth it is not permissible 

 to regard them as homogeneous, for in the case of the earth the density 

 of the surface rock averages about 2.75, while the average density of the 

 whole earth is 5.53. If we let a represent the density of the sphere w, 

 the well-known expression for it suggested by Laplace is 



G sin 



o = 



'° (''s") 



r 

 a 



(11) 



where r is the distance from the center of m, and where G and fx are con- 

 stants depending upon the constitution of the body. According to this 

 law the density of the body increases from the surface to the center, and is 

 finite at both the surface and the center. We shall determine G and n by 

 making both the surface density and the mean density agree with the 

 results furnished by observation. 



The mass is found from the equation 



/a pa 



or^dr = 4;ra'G / — sin ( u—) d — = — ?— [sin a— u cos u] (12) 

 / a \ aj a p? '^ 



Let a'^^ represent the surface density and a the mean density. Then 

 G and [i. are determined by the equations 



a'**' = (j sin // djw2 = 3(T [sin /x— j« cos j«] (13) 



The density at the center is 



<7«'^ = G/X (14) 



In the case of the earth a/"^ = 2.75, ai = 5.53, whence it is found from (13) 

 and (14) that 



(? = 4.39633 /I = 2.46579 a^^''' = 10.840 (15) 



In our ignorance as to the density of the surface material of the moon, 

 W2, we shall assume that for it //2 = ju and determine G^ from (12) so that 

 81.7^3 = mi. We find, taking a, = 3,958.2 miles and 02 = 1,081.5 miles, that 



(?2 = 2.6393 (16) 



whence 



^2 = 3.32 a2'«^ = 1.65 (T2'"^ = 6.51 (17) 



