90 



GRAVITATIONAL METHODS 



[Chap. 7 



of the earth, and to express this distribution as a simple function of lon- 

 gitude and latitude. The coefficients of the final equation may be deter- 

 mined from gravity measurements in different latitudes and longitudes, 

 leading to an empirical formula for the variation of the normal value of 

 gravity distribution at the surface. The only assumptions made in its 

 derivation are that the surface of the earth is a niveau surface, and that the 

 earth consists of concentric and coaxial shells on which arbitrary changes 

 of density may occur. Stokes and Poincar^ showed later that the theorem 

 of Clairaut follows alone from the assumption that the earth's surface is a 

 niveau surface and that it is not necessary to assume a distribution of 

 density in concentric shells. 



Referring to Fig. 7-5, consider^" the potential at the point P' with the 

 coordinates xi, yi, zi, due to a mass element dm with the coordinates x, y, 



T^X 



Fig. 7-5. Relation of outside point to mass element in spherical body. 



and 2. The distance of P' and of dm from the origin is n and r, respectively, 

 the angle between them being 7. If the distance between P' and dm is e, 

 then V = k{dm/e). Further, 



« = VU - 0:)=^ + (2/1 - yf + izi - zf ' 

 xxi -f- yyx -h 221 , 



cos 7 = 



TTx 



and 



r = VV + ? + 



^1 = Vx? + y\ + z\ . 



1* See also A. Prey, Einfuehrung in die Geophysik, p. 60 (1922). 



