Simple Pendulum after' the String has become Slack. 811 



which the arc in question is identical with the first parabolic 

 arc. Otherwise this class would include all the paths which 

 repeat. Cases of the third class occur only when the bob 

 travels upward on the vertical part of its path, and reverses 

 its motion before reaching the upper half of the circle. 

 This can occur only when the level of no velocity lies below 

 the top of the circle — that is, for a>48° ll'. The case 

 a = 90° may be thought of as a limiting case of this third class. 

 In all the cases which are periodic the paths are symmetrical 

 with respect to the vertical diameter of the circle. 



In order to find the values of a for the various special 

 cases, a study is first made of a series of paths like those 

 shown in fig. 1. From this study some idea is gained as to 



Fig. 1. 



the neighbourhood in which certain cases are to be sought. 

 A calculation of several paths in one o£ these neighbourhoods 

 gives data from which the value of a, for the case in question 

 may be found to any desired degree of accuracy. I have 

 aimed to have my values for a correct to the nearest minute. 



Fig. 2. 



I think I have found all of the periodic paths which do 

 not involve parabolas beyond the third. In the first class 

 there are the four cases shown in fig. 2. In one of these 



