366 PROCEEDINGS OF THE AMERICAN ACADEMY, 



where A, B, C are multiple quantities or numbers and \ a number. 

 An equality that is a consequence of these relations alone will be called 

 an identity. 



We assume a space containing all the points of a given discussion. 

 Points in this space are expressible as linear functions (in the matrix 

 sense) of others. In the statement of a linear problem these relations 

 between points are the only ones that occur. Hence an expression 

 will be provably (and therefore defined as) zero only when it is reduced 

 by linear relations identically to zero. 



Lety'(ai,«2 . . . «») be a rational integral function of the n points 

 Qi and 



fiaxa^ ...«„)= 



There must then (by definition) be a set of points bi . . . b^, in terms 

 of which the as are linearly expressible, such that 



/(«! . . . an) = (p(bi . . . b^) =0. 



If any of the b's are linear functions of the others we may suppose them 

 replaced so that bx . . . b^ are linearly independent. If n > m, the 

 a's (expressible in terms of a smaller number of points) must satisfy a 

 linear relation. If 7i = m and the a's are linearly independent, the 

 equations expressing them in terms of b's can be solved and the points 

 bi expressed in terms of the a's. Then, since an identity transforms 

 into an identity, 



<^(^i . . . b^) =f(ai . . . a„) = 0. 



If w < w and the a's linearly independent, the equations express- 

 ing a's in terms of b's can be solved for n b's, and since that part of 

 <}>(bi . . . &„) containing these must vanish identically, /(ai . . . a„) = 

 as before. Thus if y'(ai . . . a„) really contains the a's, these points 

 can not be linearly independent. If they are replaced by any linearly 

 independent set of points b^, the new expression, since it vanishes, must 

 vanish identically. Therefore, if a rational integral function of n points 

 vanishes, these points satisfy a linear relation, and the given function 

 reduces identically to zero when the w points are expressed as linear 

 functions of ani/ set of linearly independent points. 



To form expressions that do vanish when the points are linearly 

 related we use ordered determinants, i.e., determinants expanded like 

 ordinary determinants with all products ordered, first term being taken 

 from first column, second term from second column, etc. Thus, 



