THE DYADICS IN THREE DIMENSIONS. 



393 



We shall represent the plane by the matrix 



[ABC] = [lax a, as a,\\ (8) 



where a^ is the coefficient of a'j in the expansion of the determinant 



Linear functions of planes are defined as in the case of points. If 

 P, Q, R are any three points in the plane ABC, it can be shown as 

 in the case of a line that a number X can be found such that 



[PQR] = \[ABC]. 



Thus {ABC\ in addition to representing a plane has a magnitude. 



From four points A, B, C, D we can form a four-rowed matrix. 

 Since this matrix contains only one four-rowed determinant, we shall 

 consider it as a determinant 



(ABCD) = 



Such a matrix of one element we shall consider as a number and indi- 

 cate this by the use of the parenthesis in the symbol (ABCD), 



2. Progressive Products. The most fundamental law of multi- 

 plication is the distributive law which can be stated in the two forms 



{A + B)C = AC + BC, 

 A(B + (7) = AB -\- AC. 



The matrix [AB] has this property. For as in equation (6) it is easy 

 to show that 



[{A + B)C] - [AC] + [BC], 

 [A{B + C)] = [AB]+[AC]. 



(9) 

 (10) 



Hence we consider [AB] as a product of A and B. The process of 

 multiplication consists in placing the matrix A over the matrix B 

 to form the two-rowed matrix 



a\ 0-2 as ai 

 bi bi bs bi 



