GRAVITATIONAL METHODS 253 



the poles, because at the poles the value of :r is zero. At the equator the 

 component of centrifugal force opposing the attractional force of gravity 

 is 3.39 cm/sec.2, v^hich is an appreciable amount in the refined measure- 

 ments necessary for gravity exploration. The force of gravity therefore 

 varies with latitude. 



The Variation of Gravity with Latitude.^ — Assume the earth to be 

 a sphere with radius r, as shown in Figure 124, upon which the force of 

 gravitational attraction A is the same at all points on the surface. At a 

 point at latitude ^, ^r^the resultant of the attraction (A), and the centri- 

 fugal force (c) is found as follows : at the equator, centrifugal force Ce = 

 w^ r.* The centrifugal force at the pole Cp = 0. li ge = gravity at the 

 equator, and gp = gravity at the pole, then : 



ge = A-Ce (6) 



gp = A-Cp = A (7) 



ffp~ ffe = Ce (8) 



In latitude </>, x = r cos (/>, and the centrifugal force in latitude 4> 

 (symbol c^) is : 



C(f, = w^ r cos cf) = Ce COS (f> (9) 



since Cg = oP' r. The component of c^ opposed directly to gravity A is f^ 

 cos </) (along the radius r) where C0cos <^ = Cg cos-</), so that: 



g^ = A — Cecos^ffi (10) 



Substituting in Equation 10 the value of ^ in Equation 6: 



g<f> = ffe+ Ce—CeCOS^(f) 

 = ge+ Ce (l-COS^^) 



= ge + Ce sin^ ^ 



= 9e+ (gp - ge) sin2 (/> ( 10a) 



t 111 the following, "g" and "c" refer to gravitational and centrifugal force, respectively, per 

 unit mass. The symbols are generally used to denote the accelerations of a mass due to the respec- 

 tive forces, the forces themselves being 7)ig and mc. For unit iii, the forces and acceleration are 

 equal. (See page 255.) 



* Centrifugal force for a unit mass particle at the earth's surface at the equator is 

 C/Vr, where r is the equatorial radius and U the velocity of the particle or the distance 

 it moves, divided by the time involved. For such a particle the distance covered in one 

 day is Ztt r (the circumference of the earth). The time of rotation of the earth (T) is 

 24 hours = 86,400 seconds. From the above : U = 2 7r r/T, and f/' = 4 it' r* /T; 

 centrifugal force is 4 ir~ r/T', and in terms of angular velocity w it is wV. 



