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NOTE VI. 



PENDULUM. 



THERE are two properties of the cycloid which enable 

 us to determine, with considerable ease, the motion of a 



particle oscillating in a resist 

 ing medium, and constrained 

 to describe that curve. Let 

 C P B be any cycloid with 

 the tangent at the cusp B 

 vertical. Let C A be its 

 axis, and on it describe the 

 semicircle C Q A. Now, P being any point on the 

 cycloid, draw P Q N perpendicular to C A, cutting the 

 circle in Q, and join C Q. Then the properties referred 

 to are, 



1. The tangent at P is parallel to C Q. 



2. The arc C P is twice the chord C Q. 



A particle is constrained to move in a cycloid under the 

 action of gravity, to determine the motion. 



Let m be the mass of the particle, w its weight, and let 

 / be twice the diameter of the generating circle. Sup 

 pose P to be the position of the particle at any time t, 

 and let the arc C P be s. 



The weight of the particle may be resolved into two, 

 one along the normal and one along the tangent. The 

 effect of the former will depend on the manner in which 



