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' t 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 



a!* + tf*+(L-a??=D .................. (1). 



Now the equations of motion are, by Article 280, 

 mx' - Zmtly sml = -T (of /I), ) 



my + 2ml(x'sml + z'cosl) = -T(y'/L), I ...... (2). 



mz' - 2raH' cos I = - 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 



/=|(^+y' 2 )/i ........................ (3). 



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

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

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

 Omitting z'* in the integral equation, and substituting for z from 

 (3), we have 



\ m (x' z + ' 2 ) = const. - W (x'* + y *)/L .......... (4). 



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

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

 integration 



x'y'y'x= fl sin I (a* + 2/' 2 ) + const .......... (5). 



Introducing polar coordinates in the horizontal plane given by 

 #' = r cos 0, y' = r sin 6, 



