OF A FUNCTION OF TWO INDEPENDENT VARIABLES. 233 



234. A function u of two variables may have a maximum 

 or minimum value for values of x and y which render -y- 



and -y- indeterminate or infinite. Such exceptional cases must 



be examined specially, as there is no general theory appli- 

 cable to them. For example, suppose 



du 2x du 



Here, when x and y vanish -y- and -j- become indeter- 



dx dy 



minate. If we put y = ax, we have 



du 2 du 2q 



Hence -j- and -7- are infinite when x = 0, and y = Q. But 



c? JC ^%Y 



w is really a minimum then, for it vanishes only when x and 

 jr/ vanish and is never negative. 



235. On a case of maxima or minima values of a function 

 of two independent variables. 



If u denote a function of two independent variables a; and y, 

 the values of x and y that make u a maximum or minimum 

 are found from the two equations 



^ = 0, ^ = 0. 

 ax dy 



If these equations are satisfied by a single relation between 

 x and y, we cannot determine a finite number of values of x 

 and y, that render u a maximum or minimum. This case we 

 propose to examine. 



Suppose u = <j>(x,y) ....................... .......... (1), 



^=U.M, ^ = V.M.. ...(2), 



dx dy 



where U, V, M are functions of x and y. 



