162 PROFESSOR TA1T ON THE APPLICATION OF HAMILTON'S 



where H and fx are essentially positive, the free path is an ellipse of which the 

 origin (the centre of force) is a focus. 



This ellipse is the brachistochrone for the potential V 1 , and whole energy II l , 

 where 



2(H 1 -V 1 ) = 2 (r -11 )' 



or 



This corresponds to a central force 



2 



Or 



4(//-Hr)' 



dV 1 _ _C CHr 



dr - 4(/z-Hr) + 4(/x-Hr) 2 



_C^ 



~ 4(/x-Hr) 2 



The velocity at any point is 



/ Cr 



V '2(lL - 



2(/x-Hr)' 

 In the ellipse, we know by ordinary kinetics that 



Comparing this with the above formula (25) we have 



Hence the velocity in the free ellipse is 



v=j£j?^' ■ • ( 26 > 



x a ^ r 



That in the same ellipse, when it is a brachistochrone, is, as above, 



/ Cr jQa I r 



2(/z-Hr) ^ fx * 2a 



But if we refer it to the other focus of the ellipse we have 



r x = 2a — r . 



Hence 



Comparing (26) and (27), we have the singular result that a planet moving 

 freely about a centre of force in the focus of its elliptic orbit is describing a brachis- 

 tochrone {for the same law of velocity as regards position) about the other focus. 

 The reason of this remarkable property, as well as of the connected one that 



