GRAVITATIONAL METHODS 



307 



H is produced which tends to twist the beam and which is the resultant of 

 the two forces designated as fi and /g in the figure. 



Where / = the half length of the balance beam, the moment of force Mi 

 is given by : 



Ml = IH (41) 



Since fi and /g are approximately equal, the subscripts may be dropped and 

 the symbol / used for either of them. 



The angle between fi and /g is e, as they are perpendicular to the equi- 

 potential surfaces Uz and Ui which converge at that angle. Referring to 

 Figure 169, it is seen that 



f = m g 

 and that the horizontal projection H is approximately 



H = mg c 

 From the figure also : 



Ml = I mg € 



(42) 



(43) 



(44) 



From Equation 40 which refers to the convergence angle of equipotential 

 surfaces c, if we let h = dh; then 



c = h/g ' dg/ds 



If this value of e is substituted in the equation for Mi, that quantity can be 

 written thus : 



Ml = I mg h/g • dg/ds 



I mh dg/ds 

 (45) 



We assume that the maximum hori 

 zontal gradient of gravity is in the direc 



tion ^ and that this direction is at right J 



angles to the beam. Also, it is assumed 

 that for the position of the beam under 

 consideration its hanging weight end 

 makes the angle </> with the astronomic 

 north direction x, and that 3; represents 

 the east direction, as shown in Figure 170 

 in plan view. 



This figure illustrates the nature of the rate of change of gravity in 

 the direction s, or that 



PLAN VIEW 



Fig. 170. — Plan view of forces due to 

 the gradient of gravity acting on a tor- 

 sion balance beam. 



dg/ds = dg/dy cos <f> — dg/dx sin ^ 



(46) 



