82 HI. GENERAL PRINCIPLES. WORK AND ENERGY. 



Now 



t = 



o 



For any particular curve x is a given function of 3. Giving it a 

 variation dx we have 



x'dx'dz 



ot 



-f 



Making use of dx' = ^--- and integrating by parts 1 ), 



Zi Z l 



x' dx 



de 



If the ends of the curve are fixed dx vanishes for both limits # 

 and 0i, hence the integrated part vanishes. Consequently for a 

 minimum the integral must vanish. 



Now since the function da; is purely arbitrary if the other factor 

 of the integrand did not vanish for any points of the curve we 

 might take dx of the same sign as that factor at each point. Thus 

 the integrand would be positive everywhere and the integral would 

 not vanish, consequently the factor multiplying Sx must vanish 

 for each point of the curve, or 



This is the differential equation of the curve of quickest descent, 

 or brachistochrone. 



Integrating we have 



x' 



- = c, an arbitrary constant. 



y(l + *")(V-*0[*-*o]) 

 Squaring and solving for x t2 we obtain 



V 2 1 V 2 



Let us put a = ~ -f # , & = ^ ^ ^ ^ (6 is arbitrary, since it 



involves c), then we have 



1) The bar / signifies that we are to subtract the value of the expression 

 before it at the lower limit z from the value at the upper limit z : . 



