312 EXPLORATION GEOPHYSICS 



sented previously that the rate of change of gravity potential (7 in a given direction 

 equals the force in the opposite direction (note that the directions x' and y' are in the 

 horizontal touching plane and not in the equipotential surface). 



The rates of change of gx in the x' direction and that of g^ in the y direction are 

 given by the following expressions : 



and 



32/ 



"dx' "dy' 



also 32^_ _ 3!^/ and^^= -^^ 



'dx' 'dx'^ 3/ Zy'^ 



Heretofore we have considered only conditions along the directions of principal 

 curvature x' and y of Figure 173. It is also assumed that, for the small area occupied 

 by the torsion balance, the rate of change of gx in the y direction and the rate of 

 change of gy in the x' direction are also uniform. In words, the rate of change of the 

 N-S horizontal gravity component is uniform in the E-W direction and that of the 

 E-W component is uniform in the N-S direction. 



On this basis, establishing the center of the horizontal touching plane of Figure 173 

 as the center of coordinates, the values of gx' and of g^ for points in this plane whose 

 coordinates are distances x' and y can be found. For example, take a point A with 

 coordinates x\ y ; the value of the N-S horizontal component at point A, or 



(^/). = 1^ • ^+1^ • y (49) 



ox" oyf 



In words again, this means that the value of the N-S horizontal component at a point 

 A is the rate of change of this component in the N-S direction times the x' coordinate 

 of the point, plus the rate of change of this horizontal component in the E-W direction 

 times the y coordinate of the point. In like manner 



(^;)a = |^'- ^'+1^ • y (50) 



?) x" oy' 



The resultant horizontal gravity component at point A (Figure 174) which might 

 be one of those pictured in Figure 173 would be, introducing second derivatives, the 

 vector sum of 



{< 



Fig. 174. — ^The resultant of horizontal 

 gravity components at a point in the 

 horizontal plane touching an equipoten- 

 tial surface. Point designated as A. 



(p.,y.)A--|^^.^-f^^j^-/ + ^^^:^.^+^-y| (51) 



Where x' and y' are positive, the minus sign indicates a resultant in the negative 

 direction. This is the general equation for the change of the two horizontal gravity 

 components in the horizontal plane, assuming the change may be considered as linear 

 or uniform. 



