354 RELATIVE MOTION AND UNIVERSAL GRAVITATION. [CH. XIII. 



For the motion of the body, we have, by resolving parallel to 

 the direction of W y 



- W + (ymE/R 2 ) cos (7 - X) = mR& cos X cos I ...... (1), 



and, by resolving at right angles to the radius, 



TFsin(Z--X) = raJ [ m 2 cosXsinX ............... (2). 



Now mjRO 2 /TT is, in all latitudes, a small fraction, approxi- 

 mately equal to 1/290, and thus the angle I X is very small. 

 From equation (1) we have, when 1 is neglected altogether, 



and for a second approximation 



W=ymE/R*-mRWcos*l ............... (3). 



To this order of approximation, the weight of a body differs 

 from the force with which the Earth attracts it by the vertical 

 component of the force required to maintain the circular motion 

 about the polar axis. 



279. Deviation of the plumb-line. The latitude I and 

 the angle X, which the Earth's radius makes with the plane of 

 the equator, are connected by an equation, which is most readily 

 obtained by resolving for the motion of the body at right angles 

 to the direction of W. We find 



(ymE/R 2 ) sin (I X) = mRl 2 cos X sin I, 

 so that tan X = tan 1 1 1 -- Wrpz) ..................... (-0- 



The angle I X is the deviation of the plumb-line, i.e. the 

 angle between the vertical and a line drawn to the Earth's centre. 



The components of the force supporting the body parallel and 

 perpendicular to the polar axis are determined by resolving in 

 those directions. We find 



- W sin I + (ymE/R 2 ) sin X = 0, 



- W cos I + (jmE/R 2 ) cos X = mRtl 2 cos X,J 



so that we have 



sin X _ {(yg/g) - JSO'} cos X _ 



j ""~ 7 " fr I tit ( 2 / 



sin I cos I 



and these equations are in accord with equation (1). 



