152 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON'S 



and the central force at distance r is evidently 



dV _ nfi 

 ~ ~dr~ ~ 2r n + l ' 



Thus (2) becomes 



f^lY + (*L\ 2 + f^X - - 



\dx) \dy J \dz) "~ fx 

 or, changing to polar co-ordinates, 



\dr) + r 2 \dd) + r- 2 sin 2 \tf0/ 



It is obvious that we must take 



dr\ 2 r" 



^=0 



d<p ' 



which shows that the path is in a plane passing through the centre of force. 

 The above equation will then be satisfied by 



dT - n — — P" " r 

 dd~ ' dr~sl^-^r- 



Hence we have 



=««+/*-JF?' 



■=«»+^{^- 1 - coI, ^} +c - 



And the equation of the brachistochrone, which is evidently a plane curve, is 



2a 



I_J A*"* ■ ^ J; 



V W" 1 r * r"+ 2 



2 -i V/JLCL 



= — pr cos " + 2 : 



« +2 r 2 



!L±2 /- n+2 .. _, 



or r 2 = -v/^a sec — ^ — (#-&), 



