250 MAXIMA AND MINIMA VALUES 



are satisfied, u is a maximum or minimum ; for then (3) is 

 necessarily positive, therefore (2) has always the same sign as 

 A, and u is a maximum if A be negative, and a minimum if 

 A be positive. 



Hence the necessary and sufficient conditions for the 

 existence of a maximum or minimum value of a function u of 

 three independent variables, are, that the values of x, y, z 

 drawn from 



du = Q du = Q du = Q 



dx dy dz 



d~u d*u f cPu \ 



should make -r-s -?- -> r~ positive. 

 dar dy* \dx dy) J 



(d*u d*u tfu tfu V . 



and -j-o i T j r- -7 r less than 



\ dx dy dz dx dy dx dz) 



<Fu d*u _ r tfu V[ (d*u <j?u _ / 



_ 

 dy* dx dy } d& dz* d^dz 



It follows of course from these conditions, that 



, ,, u u u ,, i 



and thus -j z , -j- s , -v- 2 must all have the same sign, and u 

 dx dy cLz 



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

 be positive. 



From the conditions (4) and (5), we should conjecture by 

 the principle of symmetry, that BC A' 2 will also be positive 

 if (4) and (5) hold. This is easily verified, for from (5) we 

 find that 



A [ASC+ ZA'B'C' - AA'* - BE'* - CC'*} 

 is positive, and therefore, since A and B have the same sign, 



by (4) 



B [ABC + ZA'B'C' - AA* - BB' Z - CO'*} 



is positive, and therefore 



(BB' - AGJ is less than (BO- A' 2 ) (BA - C' 2 ), 

 from which it follows that BO A 3 is positive. 



