176 G. F. BECKER ISOSTASY AND RADIOACTIVITY 



that quantities as great as 10 e^ may be neglected in determining the 

 mean figure of the earth. , 



To illustrate in terms of geology the. meaning of these figures, suppose 

 a spherical batholith of peridotite embedded in the outer shell of a 

 spherical earth, so as just to reach the surface at its uppermost point, and 

 consider what must be the radius of the batholith to produce a certain 

 maximum deflection. The problem is a very simple one and has been 

 fully discussed. ^^ Taking the earth's mean density at 5.5 and the surface 

 density at 2.75, let the peridotite have a density of 3.25. Then if the 

 batholith is to produce a maximum deflection of 23", it appears that the 

 radius of the batholith must be 414 miles; that it will produce this de- 

 flection at a distance of 3 miles from its point of contact with the surface, 

 and that just above its highest point it would raise the surface of the sea 

 or of the geoid by 2 feet 2 inches. 



No geologist would be surprised at the occurrence of a batholith whose 

 greatest dimension is 8i/^ miles or at the contiguity of rocks whose densi- 

 ties differ by 0.5. Now since Stokes's postulate applies not merely to the 

 external equipotential surface, but also to the interior level surfaces of 

 the globe, it would appear that such a batholith as that described repre- 

 sents the order of magnitude of the largest heterogeneities occurring 

 anywhere in the globe. 



Possibly this agreement might be pushed a little farther. Various, 

 phenomena show that the ellipticity of the equipotential surfaces dimin- 

 ishes from the exterior of the earth to its center, and so also must e/30 

 or 10 e' If this quantity measures the heterogeneity, then this must also 

 diminish toward the center ; but it would be very unsafe to conclude from 



1* It may be well to note here the formuljc for the effects produced by a spherical 

 batholith. The proof may be found- in Thomson and Tait, Nat. Phil., sections 786 and 

 787. Let a be the radius of the earth supposed spherical and r the radius of the batho- 

 lith. Let c r be the depth of the center of the batholith, p' its density, and p the 

 density of the earth's surface, while a is the earth's mean density. If i// is the maxi- 

 mum deflection of the plumb-line due to the attraction of the batholith 



r _\/ W a c- 

 o" " 2~ p'—p * 



The elevation of the geoid over the central point of the batholith is, say, h and 



A = p'— p !:° 



a a- c a- 



In the text results are given for the very high value for >(/, 2.3' '. Far commoner, though 

 still high, would be 10'' =0.0000 4848 radi'ans. With p'--p =0.5 and c=l, this gives 

 »/a = 0.00046, and with a =: 4,000 miles, » = 1.84 miles. Then h would be nearly 5 

 inches. 



It should be observed that the internal variations in density considered in this note 

 differ essentially from the variations in external form which lead to the larger irregu- 

 larities of the geoid. The great mass of the Thibetian Mountains stands above or outside 

 of the geoid. 



