172 



GRAVITATIONAL METHODS 



[Chap. 7 



dh{l — ds/r) (g + (dg/ds) -ds). Dividing by dh and ds, = dg/ds — 

 g/r — {ds/r) ■ dg/ds, so that, by neglecting second-order terms, 



^1 = ^- = til 



ds r dh ' 



(7-476) 



which states that the gradient of gravity is proportional to the curvature 

 of the vertical or to the convergence of the equipotential surfaces. A 

 torsion balance with a sensitivity of 1 E.U., in which dh (corresponding 

 to the distance between weights) is of the order of 60 cm, can therefore 

 still detect a convergence of equipotential surfaces of the order of 1/100,000 

 arc-sec. 



In addition to the forces resulting from a convergence of equipotential 

 surfaces, the torsion balance is affected by components resulting from the 



curvature conditions of a single equipo- 

 tential surface. These surfaces, in turn, 

 are customarily defined by the maximum 

 and the minimum curvatures in two di- 

 rections at right angles to each other. 

 The torsion balance does not give these 

 curvatures independently ; it gives merely 

 their difference. One E.U. corresponds 

 to a difference in the principal curva- 

 ture radii of about 4 km. 



Fig. 7-56a illustrates the gravity field 

 for a spherical surface. Since the princi- 

 pal curvatures are equal, the horizontal 

 gravity components are equal at points 

 of equal distance from the center. Their 

 resultants point to the center and no 

 beam deflection obtains. If the curvature in section I is less than 

 in section II, as in Fig. 7-566, the gravity components in the former 

 are less than those in the latter. The resultant forces no longer 

 point to the center, and give rise to a small couple tending to move the 

 beam into the direction of least curvature. It is seen that the torsion 

 balance indicates the deviation of a level plane from spherical shape. Let 

 x' and y' (Fig. 7-57) be the directions of principal curvatures, x' making 

 the angle X with north (x direction) . Assuming a linear variation which is 

 permissible within the small dimensions of the instrument, the horizontal 

 gravity components at any point P{x, y) are 



Fig. 7-57. Horizontal forces 

 and directions of principal curva- 

 tures. 



dx 



oy 



O-Kt) 



(7-48a) 



