TEE STRENGTH OF THE EARTH'S CRUST 455 



curves of force due to the same unit mass expanded into three 

 spheres of the original dimensions and with centers 50 km. apart 

 on a horizontal Hne. One-third of the mass is therefore in each 

 sphere and the density of each is 0.033. In Case A the line join- 

 ing the centers is at right angles to the vertical section plane. 

 In Case B the line joining the centers lies in the section plane. 

 The dotted lines show Case C. In this the unit mass is expanded 

 into five spheres of the original size, each sphere possessing, there- 

 fore, one-fifth of a unit mass, and consequently a density of 0.020. 

 In this case the five centers are arranged in a horizontal plane, the 

 four outer spheres having their centers 50 km. from the center of 

 the inner sphere. 



The single sphere has a volume given by the formula V = -tR^, 



in which jR = 5okm., and a density of o.ioo. The three spheres 

 have a volume therefore of 4TrR^ and a density of 0.333. The 

 three spheres make a solid of revolution whose semipolar axis is 

 equal to 2R, equatorial radius equal to R. Upon comparing this 

 aggregate to a spheroid it is seen that the double density 0.667 of 

 the intersecting portions compensates roughly for the two re-entrant 

 zones on each side of the equator. To what regular spheroid does 

 it approximate in its proportions ? Let E be the equatorial radius 

 of the spheroid and 2E the semipolar axis. The volume will be 



Q 



-ttE^, equal to the three spheres whose volume is 47ri?^, or a single 



sphere of radius 1.44R. Solving gives E=i.i4R, 2E=2.28R. 

 The spheroid with these semiaxes is shown in broken lines in Fig. 9. 

 This, then, is a spheroid which, if the density be taken as 0.033, is 

 of exactly the same mass as the original unit sphere, or the three 

 intersecting spheres, and which approximates in distribution of 

 mass and in gravitative effect to these three spheres as shown in 

 Fig. 9. The nature of the differences will be discussed later. 



Case C shows five spheres of unit volume and of density 0.020 

 whose intersecting portions would consequently have densities of 

 0.040 and 0.060. In comparing the compound mass to an oblate 

 spheroid these intersecting portions compensate roughly for the 

 re-entrants between the spheres. The limiting dimensions of the 



