OF A FUNCTION OF TWO INDEPENDENT VARIABLES. 231 

 Also for values of x and y found from these equations, 



must preserve an invariable sign, whatever be the signs and 

 values of Dx and Dy. From this we deduce the same results 

 as in the preceding Article. 



232. There is no theoretical difficulty in finding the maxi- 

 mum or minimum value of an implicit function of two inde- 

 pendent variables, nor in finding the maximum or minimum 

 value of a variable which is connected with any number of 

 other variables by equations, when the whole number of equa- 

 tions is two less than the whole number of variables. For 

 example, suppose we have two equations 



/ t (x, y, z, u) = 0, f z (x, y,z,u}=0 ............ (1), 



involving four variables x, y, z, u, and we wish to find the 

 maximum or minimum value of u. We may eliminate one 

 of the three variables x, y, z between the two equations ; 

 suppose we eliminate z ; then we obtain one equation con- 

 necting x, y, and u ; from this we find u in terms of x and y, 

 and proceed in the ordinary way to investigate the maximum 

 or minimum value of u. Or if we wish to avoid the elimina- 

 tion we may adopt the following method : consider x and y 

 as the independent variables and differentiate the given 

 equations (1) ; thus 



.(2). 



. . 

 dx dz dx du dx 



dj\ + dj\ ^. + #L^ 

 dy dz dy du dy 



*+i lb + ** 



dx dz dx du dx 



= 

 dy dz dy du dy 



From these equations we can eliminate -, and -,-, and 



fj-ji ill/ 



find -T- and -y- ; then for a maximum or minimum value of u 

 ax dy 



