Principles of the Calculus of Variations. 47 



Thus the second equation conveys the same information as the 

 first ; and as it is satisfied by A=0, there is no need in this in- 

 stance to take it into consideration. 

 In the case of the other problem, 



S^ {i/ + \vT+f)dx=0 ) 



. Xi/p n 



Adx — %ydx + XdxV l+p 2 — d. ~7yjrf ~~ ' 



Xi/p _ ft 



Apdx = 2ypdx -{- Xpdx v 1 +p* —pd . /-.-_-— 2 — u > 



or 



Ady = 2ydy + ^^~^%- = 0. 

 Hence the second equation is immediately integrable, giving 



But as the first equation is not in like manner integrable, this 

 analytical circumstance indicates that the factor p is significant, 

 and that we cannot, as in the former problem, dispense with the 

 separate consideration of the equation Ap = 0. It will therefore 

 be proper to begin with inquiring what information this equation 

 gives, before proceeding to the equation A = 0. 

 The expression for A being 



2y+— ^ ^ 



(1+/)* (l+p*h 



let us suppose that the required line is symmetrical with respect 

 to the axis of x, or that for every positive value of y there is an 

 equal negative value corresponding to the same abscissa. Then, 



1 ds 



since (1 +jj 2 ) 2 was put for ^, which has opposite signs above 



and below the axis, s being always reckoned in the same direc- 

 tion along the curve, and since is the inverse of the 



radius of curvature, which has the same sign for the positive as 

 for the negative value of y y it follows that A changes sign with 

 the change of sign of y. Hence, as the factor p changes sign 

 in the same case, Ap has the same sign on the opposite sides of 

 the axis. Clearly also it has the same value. Consequently the 

 integration of Ady = may be considered to embrace the two 

 values of y corresponding to each value of x, and thus to satisfy 

 the supposed condition of symmetry with respect to the axis oi x. 



