144 V. OSCILLATIONS AND CYCLIC MOTIONS. 



CHAPTEK V. 



OSCILLATIONS AND CYCLIC MOTIONS. 



42. Tautochrone for Gravity. A curve along which a 

 particle will descend under the action of gravity to a fixed point 

 from a variable point in the same time is called a tautochrone curve. 

 If the particle is dropped from rest we have the equation of energy 



and the time of falling to the level z = is 







Let the length of the arc s measured from the fixed point be q>(0), 

 then 



3) t-- 



o 

 If the curve is to be a tautochrone this must be independent of # or 



Let us change the variable by putting 3 = u, then 



i 





o 

 or changing the variable back to z 9 



o 



If this is to vanish for all values of the limit # the integrand 

 must vanish, or 



4) 9/0) + W(*) = > 



which is the differential equation of the curve. Writing this 



y"() = !_ 



g>'() 2/ 



