28, 29] CALCULUS OF VARIATIONS. 77 



29. Calculus of Variations. Brachistochroue. The question 

 concerning the necessary and sufficient conditions that a line integral 



(Xdx + Ydy + Zdss) 



shall be independent of the path of integration, depending only on 

 the terminal points A and B, though purely a question of the 

 calculus, is of so great importance in various parts of mathematical 

 physics that it will he considered here. For the purpose of this 

 treatment we shall make use of the calculus of variations, which on 

 account of the great use made of it in mechanics will now he 

 briefly treated. 



In the differential calculus, we have to consider questions of 

 maxima and minima of functions. A function of one variable has 

 a maximum or minimum value at a certain value of the variable if 

 the change in the function is of the same sign for any change in 

 the variable, provided the latter change is small enough. Since if 

 f(x) and all its derivatives are continuous at x 9 



/(* + *)- f(x) + hf (x) + f" (*)+..- 



If h is small enough, the expression on the right will have the 

 sign of the first term, which will change sign with li. Accordingly 

 the necessary condition for a maximum or minimum is 



Suppose on the other hand that we change the form of the 

 function - - such a change may be made to take place gradually. 

 For instance suppose we have a curve given by the parametric 

 representation, 



* = Ji(0, y = F t (t), * = F t , 



where the F y s are any uniform and continuous functions of an 

 independent variable i. If we change the form of the jP's we shall 

 change the curve suppose we change to 



To every value of t corresponds one point on each curve, con- 

 sequently to each point on one curve corresponds a definite point on 

 the other. Such a change from one curve to the other is called a 



