PHILLIPS. — THE INDETERMINATE PRODUCT. 



3G7 



«2 ^2 



= «! ^2 — Cli Cll' 



Such a determinant has certain properties of ordinary determinants. 

 Thus, if two rows are equal, the determinant is zero ; its value is not 

 changed when a multiple of one row is added to another row ; it can be 

 expanded in terms of minors taken from first m columns. As the above 

 example shows rows and columns are not interchangeable. 



The necessary and sufficient condition that ai . . . a„ satisfy a 

 linear relation is 



ai ai . . . a-i. 



^2 (Xi 



Cli 



<^n ^n 



= 0. 



For if any expression vanishes, we have shown that the «'s are linearly 

 related. Conversely, if they satisfy a linear relation, since one row is 

 a linear function of the others, the determinant is zero. 



We represent the above determinant by the symbols [ai . . . «„] 

 or («!... a„). This may be regarded as a product of the points 

 «i . . . a„. Gibbs called it the combinatorial product because it has the 

 property (characteristic of Grassmann's combinatorial) of changing 

 sign when two of the a's are interchanged. 



Similarly we define a combinatorial product 



[/(«! 



«„) <j>(hx . . . h,^\ 



when / and <^ are any rational integral functions. The two expressions 

 are multiplied distributively and each product replaced by the sum of 

 all permutations which leave the order of the a's and the order of the 

 6's unchanged, the sign being negative when the permutation is odd. 

 Thus the combinatorial product of a^a^ and bih^ is 



+ biaia^b^ — aibia^b^ — b-^a^b^i. 

 From the preceding definition it follows that 



[(«1 . . . (In) («m+l • • ■ an)] — (^i . . . «^). 



For the left hand member expresses that every permutation of 

 «! . . . a„ is to be placed in every position among the letters of every 



