3G8 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



permutation of «ot+i ...««• That is equivalent to the right side 

 which represents all permutations of aj • . . a„ and the rule of sign is 

 the same for both. It should be noted that {aiaj)ib^ is the combina- 

 torial product of {aicii) with {l\b<^ and not of «i»2 with b-Jj^. 



Let ^1 = {ax ' ' ' (tn) ^ 0. If a- is a linear function of ai . . . «„!, i.e., if 

 X lies in the space of ai . . . a„, 



{Ax) = {ai . . . cinx) = 0. 



Hence {Ax) = is the equation of that space and we may represent 

 the space by the symbol A . 

 If 



A = {ai . . . On), B = {bi . . . b,„), 

 {AB) = {ai . . . ajh . . . b,n). 



If the points a^ . . . aj)i - • -b^ are independent, i.e., if the spaces do not 

 intersect, {AB) represents the space determined by A and B. If the 

 spaces A and B intersect, {AB) = 0. Because of this property of 

 vanishing under incidence (property characteristic of Grassmann's 

 progressive product) Gibbs called this product progressive. The pro- 

 gressive product represents our first linear process and by its vanishing 

 gives the first linear relation. 



We can now express the condition that two spaces intersect. To 

 distinguish different types or degrees of incidence we need other prod- 

 ucts. We define as the regressive product in a space of order 2> the 

 result of multiplying two functions distributively and replacing each 

 term of the result by the sum of terms gotten by permuting, as in the 

 progressive product, the first ]) letters in each term. If A = {ai . . . a„), 

 B={b, . . . bj 



(AB), 



there being p elements not zero in each row of the A 's. 



If bi . . . bj is any combination of ^'s, b^ • • ■ bi the remaining ones, 



B = ^± {b, 



hj) {K 



hi\ 



