402 MOORE AND PHILLIPS. 



II. Dyadics. 



6. The indeterminate product. Grassmann ^^ showed that 

 there are four kinds of products characterized by laws which are the 

 same for units and for any hnear functions of the units. These are 

 the algebraic, AB = BA, combinatory AB = —BA, that in which all 

 products are zero, and that in which there is no relation between the 

 products of independent units. Grassmann discussed the first two 

 in detail, but it was left to the genius of Willard Gibbs to recognize 

 the importance of the last. Because of the indefinite character of the 

 result he called it the indeterminate product. 



We represent the indeterminate product of A and B by the notation 

 AB. By definition this product obeys the following laws. 



AB-\-CD = CD + AB, 



{AB + CD) -}- EF = AB-\- {CD + EF), 

 \AB + ixAB = (X + ^)AB, 

 A{B + C) = AB + AC, 

 {A + B)C = AC + BC, 

 \AB = {\A)B = Ai\B), 

 0-AB = 0, 



where A and B are extensive quantities (points, lines, planes or com- 

 plexes) and X, /x numbers. If the factors A and B of AB are replaced 

 by equivalent expressions, and the product expanded by the above 

 laws the sum of terms obtained is said to be equal to AB. 



Gibbs called the product AB a dyad. If Ai, Ao, A3,.... An are 

 extensive quantities of the same kind (points, lines, complexes, or 

 planes) and J5i, B2, Bz,. . .Bn extensive quantities of the same kind, 

 the sum 



^= AiBi-]- A2B2 -]-.... AnBn 



is called a dyadic. The A's are called the antecedents and the B's 

 the consequents of the dyadic. 



There are two products of a dyad AB and an extensive quantity C. 

 These are 



AB-C = A[BC], 



C-AB = [CA]B. 



11 Ausdehnungslehre, chapter 2, page 33. 



