373 



the particle is constrained to keep its path. If it be sus 

 pended in a peculiar manner by a string, it will merely 

 increase the tension, and thus produce no effect on the 

 motion. The force along the tangent will tend to pull the 

 particle towards C. Since the tangent at P is parallel 

 to C Q, this resolved force will clearly be ca cos Q C A ; 

 and since the angle C Q A in the semicircle C Q A is a 

 right angle, and the arc C P is twice the chord C Q, we 



have the above force, = w ^-~ = w j. 



Hence it appears that the moving force varies as s; 

 that is, as the distance from C, measured along the arc. 

 The motion of a particle under a force varying as the 

 distance has already been investigated. The motion in 

 the present case is a particular case of that theorem ; viz., 

 that case in which the particle always moves in the direc 

 tion in which the force acts. It was shown that all par 

 ticles describe their orbits round a force = p* r in the 



same time, viz., =. Hence, in our case, the time of 



V n, 



oscillation is independent of the length of the arc de 

 scribed, and is equal to TT A /_. Calling this time T, 



V CO 



we have, 



/ lm 

 T = * A/ 



V ; 



When the arc of oscillation is circular, the preceding 

 investigation must be somewhat modified. When the arc 



o 



is considerable, the time of an oscillation cannot be found 

 in finite terms. But in all practical cases the arc bears 

 only a small ratio to the radius, and the time of a whole 

 oscillation is then found to be 



rp 



V 



B B 3 



