THE CYCLOIDAL PENDULUM 267 



the radius of the circle through P will make an angle 2 6 with 

 the vertical. 



Suppose that the circle rolls along EF until the tangent to the 

 cycloid at P makes an angle 6 + dO with the horizontal. The 

 radius at P must now make an angle 2 (9 -f- d6) with the vertical, 

 so that the circle must have rotated through an angle 2 dO. Since 

 the motion of P may be regarded as one of rotation about A, the 

 small element of path ds described by P will be given by 



ds = AP 2 d0. 



Now AP AB cos 6 = D cos 6, where D is the diameter of the 

 rolling circle. Thus 



giving, on integration, s = 2 D sin 6. 



No constant of integration is required if we agree to measure s 

 from the point at which = 0, i.e. the lowest point of the cycloid. 



The property of the cycloid is now proved, and we see that 

 equation (99) is true throughout the motion of a point which 

 describes a cycloid, this being generated by the rolling of a circle 

 of diameter D given by 



212. If the cycloid is given, the frequency k of the simple har- 

 monic motion will be k = AO"' an ^ ^ ne P er iod ^ s ~T~ J or 



Thus the motion is of the same period as that of a simple 

 pendulum of length 2Z>. 



213. The importance of cycloidal motion is as follows. It has 

 been seen that the motion of a simple pendulum is only strictly 

 simple harmonic when the amplitude is so small that it may be 

 treated as infinitesimal. For finite amplitudes the motion is not 



