MATHEMATICAL NOTE 



ail C^12 .... CLln 

 Oil a^l .... CLin 



9 



flfil dni .... Ctr 



Now consider: 



(1) All the leading constituents an, 022, 033, .... a„„; 



(2) All the minor determinants obtained by selecting any two 



rows and the two corresponding columns, for instance 



CItt dra 



(3) All the minor determinants obtained by selecting any three 

 rows and the three corresponding columns, for example 



Or 



a. 



a„ 



Or 



a. 



Or 



ttrn 



a. 



ttr, 



and so on; 

 (r) All the minor determinants obtained by selecting any r rows 

 and the r corresponding columns ; 

 and so on; 

 (n) The determinant itself. 



If the quadratic expression is a "positive definite form," i.e. 

 positive in value for all values of (^r), then all the determinants 

 in (1), (2), (3), .... (n) must be positive in value. 



If on the other hand the set of values qi, q2, . ■ ■ . qn for the 

 variables xi, X2, . . . . Xn yield a maximum, then the quadratic 

 expression in (^r) must be a "negative definite form," i.e. nega- 

 tive in value for all values of (^r). The conditions are that the 

 determinants in (1), (3), (5), (7) etc. are all negative in value, 

 while those in (2), (4), (6), (8), etc., are all positive. 



If neither of these conditions holds, then the set of values 

 a;i = Qi, X2 = q2, . . . . Xn = qn does not yield a true maximum or 



