232 



the values of -=- and -. - must be zero. Thus, more simply. 

 dx dy 



ft' ?/ //7/ 



we may put ^-=0 and -r-=0m equations (2), and then 



eliminate -j- and -7- ; the two resulting equations combined 



with (1) will determine the values of x, y, z and u, which may 

 correspond to a maximum or minimum value of u. And by 

 differentiating equations (2) with respect to x and y we can 



72 72 72 ' 



find -j z , , , , and -^ , and so settle whether u is really 



a maximum or minimum. 



Practically the solution of problems of this class is facili- 

 tated by the method of indeterminate multipliers, which is 

 explained in the following Chapter. 



233. The student will find it advantageous to illustrate 

 this Chapter by means of the Geometry of Three Dimensions. 

 If z = <f> (x, y} be the equation to a surface, to find the maxima 

 and minima values of z amounts to finding those points on 

 the surface which are at a greater or a less distance from the 



ft Z 



plane of (x, y} than adjacent points. The conditions -y- = 0, 



dz 

 and = 0, make the tangent plane at any one of the points 



in question parallel to the plane of (x, y). The interpretation 

 of the case in which B* AC= will be seen from what is 

 stated in Art. 235. 



The method given in Art. 231 admits of clear geometrical 

 illustration. Ifj for example, there be a point on the given 

 surface which is at a maximum distance from the plane of 

 ( x ) y)> then in passing from that point to an adjacent point, 

 along any curve whatever lying on the surface, we must ap- 

 proach nearer to the plane of (x, y}. Now, by combining the 

 equation z = < (x, y) with y = ^r (ic),.we obtain a curve lying 

 on the given surface, and by giving every variety of form to 

 ^ (a;) we may obtain as many curves as we please. Hence 

 we see that if we put y = ty(x], and leave the form of the 

 function ^ (#) arbitrary, we do not really break the restric- 

 tion that x and y are to be independent. 



