GRAVITATIONAL METHODS 



311 



the E-W direction. The curvature radius is large for section I. The sections 

 are at right angles to each other as before. Point for point, the horizontal 

 projections of gravity forces from section II are larger than those from 

 section I. Constructing the plan vievi^ of the resultant forces at points on 

 the horizontal surface touching this elliptical equipotential surface, in the 

 same manner as in Figure 172, it is seen that the lines of force, or the 

 direction of the resultants, do not converge on or point to the center of 

 the figure. They are now parabolas. (See part b, Figure 173.) 



./ 



vt-^Ai 



! / . ,-— ^- — /^ — i^'—t* — •■- — f ; i/i; 



(b) (a) 



Fig. 173. — Part a: vertical sections I and II at right angles to each other taken 

 through an elliptical equipotential surface, and the projections of gravity force in a 

 horizontal plane which touches the surface at one point. Section II has a greater curva- 

 ture than section I, hence larger projection components. Part b: lines of force in the 

 liorizontal plane (direction of the resultants of the projection components) which in this 

 case are parabolas and do not converge to the center of the figure. 



When a simple horizontal-beam torsion balance is brought into such a 

 force field, it is evident that there are small horizontal forces tending to 

 twist the beam into the direction of the smallest curvature. Hence, such 

 a torsion balance will measure the deviation of the equipotential surface 

 from the spherical shape. 



In Figure 173, x' is designated as the N-S direction along section I, and y' the 

 E-W direction along section II. From this figure it is seen that the horizontal com- 

 ponent of the gravity force along section I, in the x' direction and denoted by gj, 

 increases outward from the center. In a similar manner along section II, the E-W 

 component gy , increases outward along the line of the section from the center, consid- 

 ering magnitudes only. It is assumed that this change in gj and gy is uniform along 

 x' and y respectively in the small area occupied by the torsion balance. The rate of 

 change is clearly not the same for gj and for gj in the case pictured, but both changes 

 are at a uniform rate. 



The N-S horizontal gravity component gj =. 



dx' 



and the E-W horizontal gravity component gy' = 



oy' 



Minus signs denote forces in negative x' and y' directions respectively, for points of 

 positive x' and y' coordinate values, and vice-versa. This goes back to the idea pre- 



