GRAVITATIONAL METHODS 



333 



The application of this method requires that the area surrounding the sta- 

 tion site be surveyed in such manner that the levels are known accurately 

 for eight points on each of several circles having points on the vertical line 

 through the station site as centers and radii of 1.5, 3, 5, 10, 20, 30, 40, 50, 

 70 and 100 meters respectively, the eight points for each circle being located 

 in eight directions and separated by an angular distance of 45°.* 



In his first paper Schweydar t expresses the heights on the various 

 circles having the station as center in terms of a Fourier series as follows : 



C(n + bn sin a+ Cn COS a + 0?n sin 2a + Cn COS 2a + 



(92) 



where Sn is the difference in height between the station and a point on circle 

 n, and a„, bn, Cn, etc. are coefficients. 



Sn sin a da 



Cn— ~ \ "f( cos a da 



1 ('' 



dn= - \ "n sin 2a da 



TT J —w 



- i r. 



^n — 1 ^n 



(93) 



cos 2a da 

 TT ) a- 



Let** 



{r,a^ = the polar coordinates of an element of the terrain material 

 referred to the station and the north direction. 

 a = the density of the material. 

 h = the height of the center of gravity of the balance above 



the station site. 

 s = the height of the element of terrain above the station site. 



The gradient in the x direction due to the element of terrain is*** 





3GcT 



r- cos a {h — z) dadr ds 



[r^^+(h-sy] 



5/2 



(94) 



* In irregular country, the leveling is done in 16 azimuths instead of 8. 

 t W. Schweydar, Zeit. fUr Geophysik, Vol. 1, pp. 81-89 (1924). 

 ** See also Broughton Edge and Laby, loc. cit. p. 309. 

 *** In the following potential discussions the magnitude of the gravity force in a 



direction r is ^r 





The development of potential theory presented early in the 



chapter used a center of coordinates at the mass (m) producing the field, and thus the 

 gravity force was given as ^r = — , where the minus sign denoted the direction 



back toward m. In balance work theory, the center of coordinates is usually taken 

 at the balance (above the mass) and the -\-s axis is directed downward. If it is 

 remembered that the force is one of attraction and thus the direction is toward in, 

 the sign convention may be dispensed with, the gravity forces being considered as their 

 magnitudes only. 



