Chap. 7] GRAVITATIONAL METHODS 169 



The accuracy is ±0.03 arc-sec. in the meridian and 0.10 arc-sec. in the 

 prime vertical. 



Much greater accuracy is obtainable by using torsion-balance observa- 

 tions. Since the torsion balance measures the rate of change of the horizon- 

 tal gravity components in the directions of minimum and maximum curva- 

 ture, differences in the horizontal gravity components may be obtained 

 by integrating their variations with distance between stations. With 

 division by gravity, the corresponding differences in the deflections of the 

 vertical can be calculated. Since the torsion balance does not measure the 

 vertical gravity gradient and furnishes the gradients of the horizontal 

 components only in the combination d^U/dy — d U/dz , relative deter- 

 minations are possible only when deflections of the vertical are known at 

 two points, at the end of a torsion-balance traverse.^^ According to Oltay 

 a comparison of astronomic and geodetic deflection observations with 

 torsion-balance measurements in the Hungarian plain gave an accuracy of 

 3 -10"' arc-sec. per km. 



In practice the determination of deflections of the vertical from torsion- 

 balance observations is simplified when geologic bodies are essentially 

 two-dimensional. In that case, the curvature values correspond to cylin- 

 drical niveau surfaces, which are fully defined by one radius of curvature, 

 and a variation of the horizontal gravity component in only one (x') 

 direction. Then the deflection of the vertical is 



^<P.' = - [^'dx' (7-46) 



g J dxi 



.The integration is carried out numerically by using averages of curva- 

 tures between closely spaced stations or by integraphs. In this manner 

 Schleusener" obtained an accuracy of about 1-10" arc-sec. per meter of 

 horizontal distance. The maximum deflections calculated from curvature 

 values, reaching a maximum of 50 E.U., were of the order of 1 • 10~ arc-sec. 

 Such deflections cannot be detected by the astronomic-geodetic method. 

 Observations of deflections of the vertical must be carefully corrected 

 for the effects of near and distant topography. Terrain effects may be 

 calculated by the use of diagrams composed of sectors bounded by radial 

 lines and concentric circles, such as those calculated by Haj-ford and 

 Schleusener.^" For the interpretation of anomalies in the deflection of the 



" R, V. Eotvos, Verh. 15. Allg. Conf. Internat. Erdmess. (Budapest, 1908). 



" K. Oltay, Geodet. Arb. d. R. v. Eotvos Geophys. Forsch. Inst., Vol. II (Buda- 

 pest, 1927). 



^' Beitr. angew. Geophys., loc. cit. 



" John F. Hay ford, "The Figure of the Earth and Isostasy from Measurements 

 in the U. S.," U. S. Department of Commerce (1909). 



^^ Loc. cit. 



