278-280] ROTATION OF THE EARTH. 355 



*280. Motion of a free body near the Earth's surface. 



We form first the equations of motion of the body referred to 

 axes fixed in the Earth. We take the origin to be at the centre 

 of the Earth, the axis of z to be the polar axis (from South Pole to 

 North Pole), the axis of x to be the intersection of the plane of 

 the equator and the meridian plane near which the motion takes 

 place, the positive sense being from the centre to the meridian 

 in question, also we take the axis of y to be at right angles to 

 this meridian plane and directed towards the East. This system 

 is a right-handed system. By the results of Article 263, the 

 component velocities of the body parallel to these axes are not 

 x, y, z, but they are 



and the component accelerations are 



jg(*-%)-ft(jHn), ^(y + fix) + fl (x - fly), z. 



Hence the equations of motion of the body are 



m(x- 2n - H 2 #) = - (ymE/R*) cos X,j 

 m(2/ + 2ft#-n 2 2/) = 0, I 



mz = - (ymEIR*) sin X, ) 



where X is the angle which the radius drawn through the body 

 makes with the plane of the equator. Now, as the body remains 

 near a place, we may take X to be constant, and we may in the 

 terms containing H 2 , put x = R cos X and y = 0. Then, using 

 equations (2) of Article 279, we shall have 



x - 2% = - (W/m) cos I, 

 y + 2O# = 0, 



' Z = -(W/m)sml. 



Since these equations contain only differential coefficients of 

 co, y, z with respect to the time, we may, without making any 

 alteration, suppose the origin to be on the Earth's surface in the 

 latitude and longitude near which the motion takes place. We 

 shall now, taking the origin as just explained, transform to the 

 horizontal drawn southwards as axis of od ', the horizontal drawn 

 eastwards as axis of y, and the vertical drawn upwards as axis of z'. 

 We have 



x = x sin I z cos I, y' = y, z' = z sin I + x cos I 



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