Simple Pendulum after the String has become Slack, 813 



appeared that it would be easier to imagine the rest of the 

 figure, by remembering that it is symmetrical, than it would 

 be to follow the complete figure. In the third class there 

 are three cases — one in which the second parabola degene- 

 rates into a straight line, and two in which the third 

 parabola degenerates into a straight line. Two of these 

 three cases are shown in fig. 4. As to the third case, it can 



Fig. 4. 



« = 5S°3' 



a=7l*9' 



be seen from the paths shown in fig. 3 that, when the bob is 

 on the third parabolic arc, it is moving toward one side of 

 the figure for a = 57° 51', and toward the other side of the 

 figure for a— 57° 52'. One of the desired cases must 

 therefore exist for a value of a, which lies between these 

 limits. 



It is very noticeable that the values of a for which these 

 periodic cases occur are not at all uniformly distributed 

 throughout the possible ninety degrees. Their distribution 

 is indicated in fig. 5. On the three horizontal lines are 



Fiff. 5. 































1 

 1 i 



1 

















1 1 



1 511 



1 



1 



IIS 













1 



I ill 



1 



III 





20' 



40' 



60< 



80' 



shown the values of a for the cases in which there are in- 

 volved (1) only the first parabolic arc, (2) only the first two 

 parabolic arcs, (3) only the first three parabolic arcs. It 

 will be seen that, with only one exception, all of the cases 

 here considered are included between the limits of a = 50° 

 and a = 72°. 



Northampton, Mass., 

 December 23, 1920. 



Phil Mag. S. 6. Vol. 41. No. 245. May 1921. 3 II 



