under Gravity, suggested by a proposition of Galileo s. 



235 



(y — j3) and {/3 — u), the angles which the upper and lower 

 parts of the path make with the direct line OB. The results 

 are shown on the upper and lower curves of fig. 8. When 



Fiff. 



the line is horizontal both ancles are 45°, as has been alreadv 

 mentioned. For inclined positions the upper angle is the 

 greater, and as the vertical is approached this ratio approaches 

 three. Thus the intermediate point for the quickest path 

 between two points nearly on the same vertical approaches a 

 position one quarter way down the line, in other words the 

 times of describing the two parts of the path are equal. 



On examination the rather remarkable result is found 

 that this is a general property of the minimum path. The 

 condition for equality of times is c/vi = a/(v 1 + v 2 ), 



or sin(7 — /3)cosec(/3 — a) = l + ]sin ^cosecYsin (7— a) 



cosec(/3 — «)|-2\ 



If this be combined with # = 37 — 7r it will be found to lead 

 to the equation already found to connect #7. 



The method used for drawing the loci for constant times 

 can easily be modified to meet the case in which the particle 

 has an initial velocity at the upper point. The points are 

 got as the intersections of two sets of limacons. The cur\ es 

 break off from the upper point, and as the initial velocity is 

 increased it is evident they become loci for equal distances : 

 i.e. ellipses with foci at the terminal points. The tendency 

 in this direction is seen in fio-. 9, which is drawn for the case 



