29] BRACHISTOCHRONE FOR GRAVITY. 



If we introduce a new variable # such that 



I} a4-b 



g --- ~ COS ft, 



we have 



dz = -s 



83 



Thus our differential equation becomes 



dx -i/l -f cos & 1 -f- cos # 



Consequently 



Integrating, 



1 cos & 



sin # 



where <i is an arbitrary constant. 

 Combining this with 



z + 6-^(1 -COB*), 



we have the equations of the curve in terms of ft, a parameter 

 which may be eliminated from the two equations. 



If a vertical circle of radius c = A roll under a horizontal 



i 



straight line (Fig. 19), and # be the angle made with the downward 

 vertical by a radius fixed in the rolling circle, the distance moved 



Fig. 19. 



by the center of the circle from the position in which # = is 

 equal to the arc rolled over, Aft. A point at the extremity of the 

 given radius lies then at a horizontal distance J.sin^ farther, so 

 that its horizontal coordinate is 



Its vertical coordinate measured from its initial position # = is 



z = A(l cos#). 



6* 



