Chap. 7] GRAVITATIONAL METHODS 89 



the work performed by a mass of 1 g in falling from space upon the 

 earth. Since the gravity force g, in accordance with Newton's law, is 

 g = k(M/R^) (M = earth's mass, R = earth's radius), and since work is the 

 product of force and distance, the attraction potential V = kM/R = 

 6.25-10^^ ergs. The gravity potential may also be defined as potential 

 energy of the unit mass. Since the potential energy of a body of the 

 weight m-g at an elevation h is m-h g and since, at the earth's surface, 

 g = kM/R^, h = R, and V = kM/R, the potential energy is m-V. Ac- 

 tually the system to which this potential is referred is not stationary but 

 rotates with the earth; hence, the potential of the centrifugal force, or 

 V' = ^o}^(x^ + if), must be added to the attraction potential. The total 

 potential at the earth's surface is usually designated by the letter U = 

 F + y ; CO is the angular velocity of the earth's rotation, or 27r/86,164sec~ . 

 For any point outside a heavy mass, the potential function with all its 

 derivatives of arbitrary order is finite and continuous and controlled by 

 La,place's equation: 



^_^^^^_2^ = 0. (7-5) 



Points of equal value of U may be connected by "equipotential" ("level," 

 or "niveau") surfaces. The potential gradient in this surface is zero, and 

 no force component exists. Any equipotentiai surface is always at right 

 angles to the force. The value of gravity can change arbitrarily on a 

 niveau surface; hence, a niveau surface is not a surface of equal gravity. 

 The ocean surface is an equipotentiai surface of gravity, since the surface 

 of a liquid adjusts itself at right angles to the direction of gravity. The 

 distance of successive equipotentiai planes is arbitrary and depends on 

 their difference of potential. The difference in potential of two surfaces 

 1 cm apart is 980 ergs; conversely, the distance corresponding to unit 

 (1 erg) potential difference is 1/980 cm. The interval h between successive 

 planes is a constant and is inversely proportional to gravity, or C = g-h, 

 where h is the interval and g gravity. 



Fundamentally, the aim of gravitational methods is to measure "anoma- 

 lies" in the gravitational field of the earth. Since it is not possible^^ to 

 compensate the normal field by the technique of measurement (as shown in 

 Chapter 8, a compensation of the normal terrestrial field is possible in 

 magnetic instruments), its value must be computed for each point of 

 observation and must be deducted from the observed gravity. The 

 theorem of Clairaut makes it possible to calculate the normal distribution 

 of gravity from the mass and figure and the centrifugal force at the surface 



"" This applies to the total vector and its vertical component. Horizontal 

 gravity components may be compensated (as in the gravity compensator, see p. 87). 



