170 GRAVITATIONAL METHODS [Chap. 7 



vertical, graphical methods or integration machines are employed. Since 

 deflection of the vertical is horizontal gravity component divided by 

 gravity, diagrams and integraphs developed for the (vertical component 

 of) gravity are applicable with 90° rotation of geologic section or diagram. 



VIII. TORSION-BALANCE METHODS 



A. Quantities Measured; Space Geometry of Equipotential 



Surfaces 



The torsion balance measures the following physical quantities: (1) 

 the "gradient," or rate of change of gravity, related to the convergence of 

 equipotential surfaces and to the curvature of the vertical; (2) the so-called 

 "curvature values," or "horizontal directing forces," which give the 

 deviation of equipotential surfaces from spherical shape, and give the 

 direction of the minimum curvature. The curvature values represent the 

 north gradient of the east component of gravity and the difference of the 

 east gradient of the east component minus the north gradient of the north 

 component. From the previous discussion of gravity, gravity potential, 

 and surfaces of equal potential,^^ it is seen that the distance between the 

 latter is inversely proportional to the gravity. Hence, a convergence of 

 equipotential surfaces corresponds to a horizontal change or a gradient of 

 gravity. If the change between two points is uniform, g' = g -\- (dg/ds) • ds 

 where (dg/ds) is the gradient of gravity at right angles to a line of equal 

 gravity (isogam). The gradient may be resolved into a north and east 

 component so that 



dg 



dg 



ds 



=.,/riUY+(---y -^l tana=^. (7-47a) 

 y \dxdz) ^\dydzj dg 



dx 



Gradients are expressed in Eotvos units = E.U. = 10~ C.G.S. units or 

 10~^ Gals cm"\ Isogams are usually drawn at intervals of 1 or § milligal. 

 A change in gravity by 1 milligal for 1 km distance corresponds to a gra- 

 dient of about 10 E.U. If h and h', respectively, represent the spacings 

 of two equipotential surfaces with the convergence t between two points 

 whose horizontal separation is ds, then dh — dh' = dh = ids. If r is the 

 radius of curvature of the vertical, dh = t-r. Since gdh = g'dh', 

 gdh = (g + (dg/ds) ■ds)-(dh — ids). Substituting t = dh/r, gdh = 



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