OF A FUNCTION OF SEVERAL VARIABLES. 253 



to be quite arbitrary, it follows, in the same manner as in 

 Art. 232, that if </> is to be a maximum or minimum, we 

 must have 



Z=0, F=0, Z=0, # = (3). 



From these m n equations, combined with the n given 

 equations, we can find the values of the variables for which 

 <f> may be a maximum or minimum. To determine whether 



< is a maximum or minimum we must express -33?* From 



ctt 



(4), with the use of (5), we have 



We should then examine whether the above expression 

 retains an invariable sign, when the specific values of the 

 variables x, y, z, ... are used, whatever be the arbitrary 

 values assigned to Dx, Dy, Dz, .... If it does, then </> is 

 a maximum if that sign be negative, and a minimum if it 

 be positive. 



239. The practical solution of any example according to 

 tne above theory is facilitated by making use of indeterminate 

 multipliers. Multiply the first of equations (3) by \, the 

 second by X 2 , ... the n ih by X n , the values of X,, X a , ... X n 

 being at present undetermined. Add the results to (2), then 

 we may write 



dt \dx 1 dx 2 dx 3 dx 



. 



1 dz 2 dz * dz 

 + (6). 



