156 PROFESSOR TAIT ON THE APPLICATION OF HAMILTON'S 



But d (dr \ _ <Pt dx d 2 T dy d 2 r dz 



dt\dx) ~ dx 1 dt dydx dt dzdx dt 



_^^d^d^ d 2 r dr d^dr\ J^ dM 

 ~®{dx 2 dx + dydx dy + dzdxdz ) " 2® dx ' ™'' 



which is the ordinary form of the equation of the brachistochrone, (A) in § 4. 



Also, dM _ 2 f dT d Sdr\ drd_ /dr\ dr d /dr\ \ 



dt ~ \dx dt \dx) dy dt \dy) + dz dt \dz) f 



IT dr_<ffl , dj^M dr <ffl) 

 ~W\dxdx + dy~dy + dzdz) ( 12 )" 



d 2 x 



The above value of -r^ becomes therefore 



dr 

 <Px _ J^ d& _ jr- f dr d® dr_ m dr ffl, \ 



dt 2 ~ 2W dx — \dxdz + dydy + dzdz$ ' ' ' (13) ' 

 which (8) reduces to the form 



dr 

 d 2 x v , n -;— ( „ dr ^ T dr „ dr ) 



aT = -X + 2*{x s - + Y^ + Z^-} . . (14). 



And we have, of course, similar expressions for -J[ and -A-. 



13. We may thus easily prove the fundamental property of brachistochrones 

 given in most treatises on dynamics. 



The pressure on the curve, due to the motion, is equal to that due to the impressed 

 forces. 



For (14) may be written 



ff = -X + 2^®{x^+Y^ + Z^l 

 dt 2 dt \ dt dt dt \ 



— _X + 2— JX— + Y^-+ Z — ^ 

 ds \ ds ds ds J 



=X-2JX- — (X — + Y^-+ Z — \ l- 

 \ ds \ ds ds ds J J 



Now X ;^-+ Y 77^+ Z ;r" * s tne component of the impressed forces along ds. 



Hence 



v dx / v dx v dy „ dz\ 



Y_^(X^ + Y^+Z^V Z-^(X^+Y^ + Z^V 



ds \ ds ds ds J ds \ ds ds ds / 



