150 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON'S 



Hence 



(d,T\ 2 /^ T V f^ r \ — ^ 



doc) + \dy) + \dz) ~ 2g(a + yy 



This equation is obviously satisfied by 



/dr\ doe 



\^2/ (ft 



Hence -^= ™ that is the path is in a vertical plane. We may take this as the 

 plane of xy. Hence our equation becomes 



sdry + (dr.\ 2 .= 1 . 



Vote/ \dyJ 2g(a + y) 



We may now write 



dT _■ 1 



dx \/2gb 



\dy t 2g\a + y bJ, 



(5). 



where b is an arbitrary constant. 

 By (5) we have, at once, 



V2(f r = -£ + fay J— 1 



Hence the equation of the brachistochrone is (by § 6) 



(6). 



dr 



— = const. 



db 



or C 



' a + y b 



- _ £L J_ 1 / * ^ 



6t + 6 2 7 r~r~ _ r ' 



that is, changing the constant, and effecting the integration, 



°i = - * -n/(» ~ ^) (« + 2/) + | vers^ ?£ + ») (7 ). 



the common equation of the Cycloid, the velocity at any point being that due to 

 a fall from the base. 



In this case we have evidently 



