242 NEWTON S PRINCIPIA. 



follows for variations of velocity the same law that any 

 element of the pendulum experiences, but whose magni 

 tude is such that the whole moments of the resistance on 

 the pendulum and particle about the axis of suspension 

 are equal. In reasoning then on the pendulum we may 

 always consider it as a simple pendulum. For when we 

 have determined its motion, the preceding formulas enable 

 us to determine that of the compound pendulum. 



Newton does not confine himself to the case in which 

 the particle describes a circle. Supposing the string 

 flexible, he has given in the first book a way of making it 

 describe any given curve. Of all these the cycloid is that 

 which possesses the most important properties. The motion 

 of a particle in a cycloid is discussed, because it gives us a 

 deeper insight into the laws of pendulous movements than 

 that in any other curve. The motion of a particle in a circle 

 is considered, because in practice most pendulums are so 

 constructed that any point in them describes a circle. 

 [See NOTE VI.] 



A particle constrained to move in a cycloid ivhose axis is 

 vertical is acted on by gravity and resisted by a constant 

 force. To determine the motion. Newt. xxv. 



Let / be twice the radius of the generating circle, s the 

 distance of the particle at any time t from the lowest point 

 (C) of the cycloid, v the velocity, and m the mass of the 

 particle. Letjfbe the constant resistance. 



Then the moving force along the tangent is 



I 



supposing the particle to be descending. 



