254 MAXIMA AND MINIMA VALUES 



If we equate the coefficients of n of the quantities Dx, 

 Dy,... to zero, we shall arrive at n equations for determining 

 \, \, ... \ n . Substitute these values of \, \,... X n , in the 



remaining terms of (6), and - takes the form given in (4) ; 



dt 



we must therefore equate to zero the coefficients of the re- 

 maining m n of the quantities Dx, Dy, ... Hence we have 

 the rule : " Equate to zero the coefficients of every one of the 

 quantities Dx, Dy, ... in (6) ; the m equations thus found, 

 together with the n given equations, will enable us to elimi- 

 nate the n quantities \, \ 2 ... X n , and to find the values of 

 the quantities a, y, z..." 



240. The concluding part of the theory in Art. 238, in 



72 I 



which we are directed to examine the sign of , frequently 



becomes in practice excessively complicated. In fact the 

 examples of this method are generally such as allow us to 

 predict that a maximum or minimum must exist, and to dis- 

 pense with the second part of the investigation. 



EXAMPLES. 

 1. Find the maximum or minimum value of 



subject to the conditions 



ax + by + cz I = 0, 

 ax + b'y + c'z l = Q, 



Putting < for a? + y* + z*, we have 



dt 

 Also from equations (1), 



aDx + bDy+cDz = 0, 



Hence, multiplying equations (2) by \ and X 2 respectively, 

 we may put 



