100 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON S 



Here it is obvious, by (18), that (t) is the action in a. free path coinciding with 

 the brachistochrone, and that 2(H X — Y x ) is the square of the velocity in this path. 

 Hence the curious result that, if r be the time through any arc of a given 

 brachistochrone, the same path mill be described freely under the action of forces 

 whose potential is V a , where 



2(Hl ~ Vl) = 2(H - V)' 



0' being any function whatever ; and (p(r) representing the action in the free path. 



17. The simplest supposition we can make is that </>'(t) is constant. In this 

 case the velocity in the free path is inversely proportional to that in the brachis- 

 tochrone at the same point ; and the action in the one is proportional to the 

 time in the other. In fact, as Professor W. Thomson has pointed out to me, in 

 this case the investigation may be made with extreme simplicity, thus — 



In the brachistochrone we have 



/ 



ds . . 



— a minimum. 



V 



Putting v = - , and considering v as the velocity in the same path due to another 

 (easily determinable) potential ; we must have 



/■ 



vds a minimum. 



This is the ordinary condition of Least Action, and belongs, therefore, to a free 

 path. 



Hence, since the cycloid is the brachistochrone for gravity, and since in it 



v 2 = 2gy, it will be a free path if v 2 = „ — , that is for a system of force where 



the potential is found from 



H — V — — 



1 l ~*w 



This gives 



da; dy fyy 



In other words, a cycloid may be described freely under the action of a force 

 towards, and inversely as the square of the distance from, the base; and the 

 velocity at any point will be the reciprocal of that in the same cycloid when it is 

 the common brachistochrone. 



This result is easily verified by a direct process. 



18. But we have, by § 16, an infinite number of other systems of forces under 

 which this cycloid will be described freely. 



