Chap. 7] 



GRAVITATIONAL METHODS 



255 



and (7-91a). In polar coordinates, x = p cos a, y = p sin a, p = r cos <p, 

 and 2 = r sin <p, so that 



U„ = 3kdm "^^ = I kdm ^^^ 



r^ 2 



[7a = —Zkdm 



p cos 2a 



= —3kdm 



cos^ ^ cos 2a 



(7-92a) 



2[/i„ = Sfcdw 



p^ sin 2a 



= 3kdm 



cos^ (p sin 2a 



Sndinh 5 4 3 2 1 t 2 3 4 5 mitrs Cunntuns 



Fig. 7-96a. Effect of three-dimensional mass on gradients and curvature values 



(after Jung). 



The following conclusions may be derived from these equations: The 

 torsion balance anomalies of three-dimensional masses are (1) inversely pro- 

 portional to the cube of the distance, other things (horizontal and vertical 

 azimuths) being equal; (2) proportional to the single horizontal azimuth 

 for gradients and the double horizontal azimuth for curvatures; (3) pro- 

 portional to the sine of the double vertical angle for gradients and the 

 square of the cosine of the single vertical angle for curvatures. By 

 comparison with eqs. (7-91a) and (7-91c) it is noted that for two-dimen- 

 sional features the effects are inversely proportional to the square of the 

 distance and to the sine and cosine of the double vertical angle. For the 

 anomaly of a sphere, the same formulas ([7-66], [7-91a], and [7-92a]) 

 apply by substitution of the total mass, M, for the differential mass, dm. 



