OF AN IMPLICIT FUNCTION. 



to zero. Or, more simply, we may put -r- = in these equa- 

 tions, and then eliminate -^ and -7- from the resulting equa- 

 tions which are 



dF dF dy dF dz ' 

 ~j -- 1 ; > 1 7 -- y- = 

 ax ay ax dz ax 



dx dy dx dz dx 



(2)v 



dx dy dx dz dx 



The equation obtained by eliminating - and -7- , com- 



doc ctsc 



bined with the equations F= 0, F t = 0, F 2 = 0, will determine 

 x, v, z and u. 



72 



By differentiating equations (1) again, we can obtain -^ , 



and by the sign which the values of x, y, z, u, already found, 

 give to this quantity, we determine whether u is a maximum 

 or minimum. 



220. Suppose we have a function of n variables, the 

 variables being connected by n 1 equations, and we require 

 the maximum or minimum value of the function. For ex- 

 ample, suppose three equations 



F(x, y, z, u) = 0, F 1 (x, y, z, u] = 0, F z (x, y, z, u} = 0, 



and that we wish to find the maximum or minimum of 

 f(x, y, z, u}. In this case, to the equations (1) of the pre- 



fill 



ceding Article, in which -7- must not be supposed zero, we 

 must add 



dx dy dx dz dx du dx 



From these four equations we must eliminate ' , -r . 



du 

 and -=- . The resulting equation combined with the given 



