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



We thus obtain the equations 



^ + 2O ( sin Z + z cos = 0, ............... (1), 



these equations determine the motion of the body relative to the 

 axes at the place of observation. 



*281. Acceleration due to gravity. Suppose the body to 

 fall from rest relative to the Earth. Then the initial velocities 

 relative to the axes at the place of observation are 



and we shall suppose that the initial value of the coordinate y is 

 zero. The motion is determined by equations (1) of Article 280. 

 Integrating the first of these, we have 



# =2ny sin (1), 



and integrating the third equation, we have 



where t is the time from the beginning of the motion. Substitut 

 ing in the second equation, and neglecting fiy, we have, on 

 integration 



so that ?/ = %l(W/m)t 3 cosl .................. (3). 



Substituting in equations (1) and (2), and neglecting terms of the 

 same order as before, we have, on integration, 



where # and z are the initial values of x and z . 



In the beginning the motion of the acceleration relative to 

 axes fixed on the Earth is directed vertically downwards and is 

 precisely ( TF/m), or we have 



and, to the order of approximation here adopted, this equation 

 holds throughout the motion. 



