GRAVITATIONAL METHODS 



315 



merits the instrument is oriented and field data on gravity forces acting are measured 

 with respect to the x and y (or the astronomic north-south and east-west) directions. 

 For this reason, it is necessary to transform the horizontal gravity components gx' 

 and gv into gravity components gx and gy, using the angle \ between these two 

 coordinate systems. 



After several transformations, which are omitted. Equation 55 becomes 



=U d'U ( \ \\ o 



-^ = g I 1 cos 2 



'^ 3y \py P' ) 



mi 



and 



(57) 



(58) 



dx' 



df 



The quantity 



d x2iy 



occurred previously following Equation 53. In the x' and y' 



system of coordinates this quantity, which represents the N-S gradient of the E-W 

 horizontal gravity component, was zero ; however, in the x, y system it has a finite value. 

 Using the shorter form of notation 



tan 2 \ = — 



2Ux 



(59) 



where Uzv = 



d^U 



, and 2 Uxy is the E-W component of the curvature quantity; 



d xdy 

 [/a is the N-S component of that quantity. 



The magnitude of the curvature quantity or R-line 

 value and the angle 2\ may be obtained graphically, as 

 shown in Figure 177. In equation form, it is : 



|R| =^-(2u:,r+i-u^y 



(60) 



Moment Due to Curvature Forces. — In this 

 discussion of the action of forces due to curvature 

 conditions of equipotential surfaces, a simple type of 

 beam is considered. It is assumed that the form 

 of the torsion balance has been modified, and the 

 masses m are placed on the ends of the horizontal 

 beam. For a 2-beam this would be equivalent to 

 moving the upper mass down a distance of h/2 and 

 moving the lower mass up a like distance. This is 

 the same as considering the action of horizontal 



gravity components in a horizontal plane acting through the center of grav- 

 ity of the beam or suspended system. Such a physical beam system is called 

 a curvature variometer and with it the curvature quantity only can be 

 measured. The horizontal components of gravity force arising from the 

 curvature of equipotential surfaces act on the horizontal part of the beam 

 of a torsion balance. 



Fig. 177.— Graphical 

 representation of the R- 

 line quantity, as obtained 

 from plotting its compo- 

 nents. Note that the U^ 

 component plots — to the 

 north. The angle 2\ is 

 shown as measured clock- 

 wise from the true north. 



