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



and putting g for W/m, we obtain from equations (4) and (5), 

 equations of the form 



and, if we put 



+ IlsmZ = < ........................ (6), 



we shall have 



r 2 + r 2 ^ 2 = (A + 203 sin 1) - r* {(g/L) + ft 2 sin 2 1}, 



,\ 



r 



These equations completely determine the motion. It is to be 

 noticed that r and < are polar coordinates referred to an initial 

 line rotating about the vertical from East to West with an angular 

 velocity O sin I, viz., with the component of the angular velocity 

 of diurnal rotation with which the stars rotate about the vertical 

 relative to the Earth. Thus the initial line from which </> is 

 measured remains parallel to a plane fixed with reference to the 

 stars. 



*283. Foucault's Pendulum. When the pendulum can 

 turn freely about its point of support and is set oscillating so as 

 to pass through its equilibrium position, the system is known as a 

 Foucault's Pendulum. 



Since r can vanish, it follows by the second of equations (7) of 

 the last Article that B must vanish, and thus < vanishes through- 

 out the motion. Hence the pendulum oscillates so that its plane 

 of vibration is fixed in direction relatively to the stars, and this 

 plane turns round the vertical relatively to the Earth with angular 

 velocity O sin I from East to West. 



The first of equations (7) of the last Article then becomes, on 

 neglecting O 2 sin 2 1 in comparison with g/L, 



showing that the horizontal motion in the plane of vibration is 

 simple harmonic motion of period 27r\/(L/g). 



If a is the amplitude of the simple harmonic motion, so that 

 the pendulum has no velocity in the plane of vibration when 

 r = a, it will not move as here described unless its angular velocity 

 relative to the Earth is fl sin I from East to West. To start the 

 pendulum, therefore, it is not sufficient to hold it aside from its 





