236 Prof. Morton and Mr. Tobin on Times of Descent 



in which the level o£ zero velocity is at a height above the 

 upper point equal to the difference of level between upper 

 Fig. 9. 



and lower points. (The times marked on the curves are given 

 as multiples of the time of fall through this vertical distance, 

 and not, as in the former set, through the length of the line.) 

 The investigation of the path for minimum time in this 

 more general case has proved intractable. It is easy to show 

 that the times along the two lines are not in general equal, 

 although this holds in the limiting case when the initial 

 velocity is very great. The intermediate point then 

 approaches the middle point of the line. 



(3) We now come to the comparison of the times of descent 

 along arc and chord when the particle starts with initial 

 velocity. It is convenient to take separately the cases where 

 the level of zero velocity is below and above the highest 

 point of the circle, Ji<2a and h>2a, A being the height 

 of the starting level above the lowest point and a the radius 

 of the circle. 



In the former case put /i = 2asin 2 7, then 7 has the same 

 meaning as in the first note, viz. the inclination of OC, 

 the chord from the lowest point to the intersection of 

 the circle with the starting level, /3 is again the inclination 

 of the chord OB along which the descent takes place. The 

 times are expressible in terms of an auxiliary angle $, defined 

 by sin </) = sin /S/sin 7, 



arc-time = v / (a/r/)F(c/>) mod. sin 7, 

 chord-time = s /(ajg) . 2 tan ^<p. 



