182 MOTION UNDER CONSTRAINTS AND RESISTANCES. [CHAP. X. 

 The period of a complete revolution is 



IT 



dd&amp;gt; 



yJo 



*193. Limiting case. In the case where the pendulum is 

 projected from the position of equilibrium with velocity equal to 

 that due to falling from the highest point the equation can be 

 integrated by logarithms. 



The constant in the energy equation of Article 188 must then 

 be chosen so that 6 vanishes when 6 = TT, and the equation there 

 fore is 



= 0(1+0080), 



which may be written 



The time of describing an angle is therefore t, where 



? : / l 



i-Vi 



i r 2 dx 11 , ( e 



sec - 

 9 



It is to be noted that the particle approaches the highest 

 point indefinitely, but does not reach it in any finite time. 



The same equations may be used to describe the motion of the 

 particle from a position indefinitely close to the unstable position 

 of equilibrium at the highest point of the circle. 



*194. Examples. 



1. Prove that the time of a finite oscillation when the fourth power of a, 

 the angle of oscillation, is neglected, is 2jr(l + ^a 2 ) J(l/g). 



2. Prove that, in the limiting case of Article 193, 



= 2 tan ~ * sinh {tj(ff/ty. 



3. Prove that if a seconds pendulum makes a complete finite oscillation 

 in four seconds the angle a is about 160. 



195. Motion on a smooth plane curve under any 

 positional forces. Suppose that a particle of mass m is con 

 strained to move on a given smooth plane curve under the action 

 of given forces in the plane. Let s be the arc of the curve 

 measured from some point of the curve up to the position of 



