618 HARMON LEWIS 



was assumed uniform. The argument that, if M < i, h x is probably 

 less than the most probable depth assuming M = i would hold 

 for the Chamberlin compensation or for a compensation uniformly 

 varying from a maximum at the surface to zero at the depth of 

 compensation. For the reduction factors in the subsidiary investi- 

 gations are linear functions of the reduction factor for uniform 

 distribution. 



Changes in formulae required by shallow depth of compensation. — 

 Under this head the approximation mentioned above in connection 

 with equation (4) will be considered. 



8hM=-8 1 (h I +h) (4) 



S x is such a quantity that (W A — « S i Vh+h]) = [W A ] M=0 is the 

 weight which column A would have if there were no isostatic com- 

 pensation. Therefore 



W A = [W A ] M=0 +aS 1 (h l +h) . 



Thus the deflection due to W A may be obtained by adding to the 

 deflection which would be produced if there were no isostatic com- 

 pensation the deflection which would be produced by aK{hA-1i). 

 Therefore the deflection at any station assuming isostatic compen- 

 sation is equal to the topographic deflection (D) plus the deflection 

 (D c ) due to the defect or excess of density from the surface down to 

 the depth of compensation. If H be the height of the observing 

 station above sea level, then the deflection due to the defect of 

 density in a compartment whose surface is h miles above sea level 

 is (neglecting the curvature of the earth), 1 



A = 12. 44 T(sma '-sinflt) j (H+hi)log =- 



A ( r r +V r 1 +(H+h I ) 2 



+ {h _ mog ^±V(^t^ [ (8) 



ri +VrS+(h-Hy ) 



In calculating the topographic deflection Hayford neglects H 

 except when it is necessary to make a slope correction. However, 

 H is introduced in equation (8) in order to make it exact. Whether 

 it would be legitimate to neglect this factor or not can best be told 



1 See A. R. Clarke, Geodesy, 1880, p. 295, and Figure of the Earth and Isostasy, 

 1909, p. 20, for the derivation of equation (8). 



