230 MAXIMA AND MINIMA VALUES 



In order that u may be a maximum or minimum, we must 

 have, by Art. 211, 



du 



therefore (-} + (^} +'(x) = 0. 



\dxj \dyj 7 



Hence, since y is really independent of x, this equation must 

 hold whatever be the function i//(#) ; 



(du\ (du\ 



therefore -j- = 0, ( = 0. 



\dxj \dyj 



In order that u may be a maximum, the values of x and y 



d 2 u 

 derived from the last equations must make -y- 2 negative, 



whatever ty'(x) ma y be > hence, denoting by A, S, C, the 



, . , (d*u\ / d s u \ , fd 2 u\ A . . 



values which , , and , respectively assume 



for the values of x and y under consideration, we require that 



should be always negative, whatever ^r'(x} may be. Hence 

 as in Art. 228, A must be negative, and generally AC B* 

 must be positive. Similarly, that u may be a minimum we 

 must have A positive, and generally AC W positive. 



The preceding method may be rendered more symmetrical 

 by supposing both x and y functions of a third variable t. 



Putting for shortness Dx for -*- , and Dy for -j- , we have 

 du du\ du 



/du\ dDx fdu\ dDy 

 \dx) dt \dy) ~dT 

 Hence we must have 



(IH- 



