Chap. 7] 



GRAVITATIONAL METHODS 



243 



sections appreciably. In the two planes of symmetry of a rectangular 

 section the gradients are zero. However, the curvatures cannot be made 

 zero. The effects of a tunnel of rectangular section, infinite in its longi- 

 tudinal direction, on gradients and curvatures follow directly from the 

 formulas for a vertical step given in the section on interpretation. Figs. 

 7-88 and 7-89 (after Meisser"''^) 

 show the gradient and curvature 

 distribution inside a tunnel which 

 is infinite in the y' direction. 

 With the notation of Fig. 7-90, 

 Uy'z' and 2Ux'v' = and 



U'^ = 2k6(a + 0) 





} (7-89) 



if the center of gravity of the 

 torsion balance is below the 

 center of the section. When 

 setting up an instrument in the 

 center line of the tunnel so that 

 its height above the tunnel floor 

 can be varied (i.e. along the z 

 axis), x' = 0, Ux'^ = 0, and 

 Ua = 8k8 tan~^ h/a. 



If the tunnel section is not 

 rectangular or nearly rectangular 

 in section, it is advisable to use 

 the graphical or integraph meth- 

 ods given in the interpretation 

 section for two-dimensional fea- 

 tures and to calculate the effect 

 of the actual tunnel outline. In 

 any event, the instrument should 

 be set up as nearly in the center 

 of the section as possible. Al- 

 though the curvatures are not 

 zero, their variation with location is least and the correction (if curva- 

 tures are used at all) may be determined with fair accurac3^ The tunnel 

 outline can be measured with a device used in excavation, sometimes 



Fig. 7-89. Lines of equal curvatures in 

 infinite rectangular tunnel section (after 

 Meisser). 



•o'Zeit. Geophys., 6(1), 17-18 (1930). 



