318 EXPLORATION GEOPHYSICS 



Combining Equations 55 and 57 gives 



( \ \\_ (^u_ _ mj\ _ (^u_ _ ^u\ 1 fggv 



^\py' P' ) \9^" ^y"/ \S^ 9/ /cos 2 X ^ ^ 



ig the last term in Equation 66 into Equation 65 produces : 



Mi = I l^^-'^] (J^ sin 2^ - H cos 2^ tan 2 \) (67) 



yd XT df J 



From Equation 58, 



dxdy 

 tan 2 X = 





Therefore ; 



Using abbreviated notation : 



Mi^Yil {U^ sin 2<^ + 2^/^^ cos 2«#.) (69) 



The above is the second part of the fundamental equation of the torsion 

 balance. The torsion balance, therefore, measures two components of the 

 curvature quantity (the N-S component and the E-W component) and 

 gives the angle A which is the direction of the section of minimum curvature 

 of the equipotential surface with the north direction. 



Units of Measurement of Curvature Values. — The unit of measure- 

 ment for the differential curvature, [ J , strictly speaking, would 



\pi P2 / 

 be in units of 1 X 10~^^ radians per cm. The significance of this magni- 

 tude lies in its indication of the difference in curvature of the two principal 

 sections of the equipotential surface, at right angles to each other. The 

 R line value used and plotted, however, is gravity (g) times this difference 



in curvature or R = g (— ^ J (see Equations 56 and 57). R is mea- 



sured in Eotvos units in terms of 1 X 10~^ dynes, since 



^"(l^^~l^) '■ ^ = 103 (10-12) = io-« 



