316 



EXPLORATION GEOPHYSICS 



Figure 173 indicates that when such a modified form of beam is placed 

 in a horizontal plane touching an elHptical equipotential surface, forces 

 act to twist the beam into the direction of the section of minimum curva- 

 ture. Figure 178 shows in plan view this type of beam placed in the 

 force field pictured in Figure 176. The angle which the beam makes with 

 the north direction is called cf>, and its angle with the x' (or minimum curv- 

 ature) direction is [x; I = the ^ length of the beam. In Figure 178 the 

 beam is at rest, the gravity force components being balanced by the torsion 

 in the suspension wire. 



Fig. 178. — Showing a simple torsion balance 

 beam (no hanging weight) placed in a horizontal 

 plane touching an elliptical eqtupotential surface at 

 one point. Gravity force components tend to twist 

 the beam into the direction of the section of the 

 surface of minimum curvature of x'. 



As seen from Figure 178, a horizontal component of force F acting 

 at right angles to the beam is produced, which is the resultant of . and 



— ^y- ■ Since a similar force F acts on both ends of the beam and in oppo- 



dy 



site directions, the curvature moment M2 is given by : 



M2 = 2mlF (61) 



The force F is derived by transforming the values of gj and g/ in the x', y' system 

 of coordinates into a system which makes the angle fi with the x', y system. This 

 latter has its .r-direction along the beam and its y-direction at right angles to the beam. 



F = gy cos II — Qx sin /t 



I, 'dU -dU ■ 



Fz=— -^ cos IX — —— sm M 

 Zy ox* 



(62) 



from Figure 179. Since g^ = —~- and g^' — —^ , the value of F may be written 



for positive x' and y', 



ay 



?)x' 



