280-282] ROTATION OF THE EARTH. 357 



It appears that the body falls a little to the East of the starting 

 point, the eastward deviation in a fall through a height h being 

 very approximately 



This result accords well with observed facts. 



*282. Motion of a Pendulum. Suppose that a simple 

 circular pendulum of length L is free to move about its point of 

 support which is fixed relatively to the Earth, and let T be the 

 tension of the suspending fibre. 



Taking x, y , z to be the coordinates of the bob referred to 

 the system of axes described in Article 280, the origin being at 

 the equilibrium position, the line of action of T makes with the 

 axes angles whose cosines are 



-x jL, -y lL, (L-z )IL, 

 and we have the relation 



x * + y *+(L-zy=D .................. (1). 



Now the equations of motion are, by Article 280, 

 mx - 2mly sml = -T (x \L\ j 



my+2ml(x sml + z cosl) = -T(y /L), I ...... (2). 



mz - Zmtly cosl = -W +T(L- z )/L. J 

 We shall integrate these equations on the assumption that 

 the pendulum makes small oscillations. On this assumption we 

 have approximately 



* =i(* 2 + 2/ 2 )/ ........................ (3). 



Multiply the equations (2) in order by x, y , z, and add. The 

 terms containing T vanish identically by (1), the terms containing 

 H also vanish identically, and the equation can be integrated. 

 Omitting ,z 2 in the integral equation, and substituting for z from 

 (3), we have 



\m (x 1 * + y *) = const. - 1 W (x * + y ^jL ......... (4). 



Again, multiplying the first of equations (2) by y , and the 

 second by x, adding, and omitting the term in y z , we have on 

 integration 



afy -y x=- O sin I (# 2 + y&quot;&amp;gt;) + const .......... (5). 



Introducing polar coordinates in the horizontal plane given by 



