CHARACTERISTIC FUNCTION TO SPECIAL CASES OF CONSTRAINT. 161 

 For by § 8 we have, putting a = 0, since the base is now the axis of x, 



= JL _ J b C0 "sV| + Jl Vb=y~+ C. 

 Hence, whatever be <p', the cycloid is a free path for the system 



\<t>' (^ b - jtcoiJi+^Vb^+c)]* 



v > = 2^-v t ) = ^-^ Wy L-. 



19. The converse of the proposition in § 16 is also curious. Taking Hamil- 

 ton's equation (18), we have, 



(0'(A)y{(0 + (fy + (f) a }= 2 (H-v)(« A ))» 



• (23). 



Comparing this with (2), we see that t = <p (A) is the brachistochronic expres- 

 sion for the time in a path which is a free path for potential V. The requisite 

 potential is now found from 



^- ir ,=2(H-V)(^(A)) 2 ... . (24). 



2(H X - VJ 



Hence, if A be the action in a given free path, the same path mill be a brachis- 

 tochrone for forces whose potential is V x , determined by (24), V being the potential 

 in the free path. 



Thus, the parabola 



(,v — &) 2 = 4a (y — a) 



is the free path for v i =2gy. And the action is given by 



1 2 1 



A = xja + ~( y - a y. 



V2g v 3 



Hence this parabola is the brachistochrone for 



1 



2(H 1 -V 1 ) = 



2Mtf>'(A)) 2 

 In the simplest case (p'(A) = 1, and we have 



_dy 1 _ _^Yi_ L 



cfcc ~ ' dy ~ kgy 2 ' 



Hence, by § 17, the parabola is a brachistochrone when a cycloid is the free path. 

 20. Again, if 



* 2 = 2(£-h), . . . (25, 



