394 MOORE AND PHILLIPS. 



and from this determining the elements of the matrix [AB]. From 

 the definition it is clear that 



\AB] = - [BA], (11) 



[AA] = 0. (12) 



Similarly the matrix [ABC] can be regarded as a product of [AB] 

 and C, of A and [BC] or of A, B, and C, the process of multiplication 

 consisting always in forming the matrix 



and determining the matrix [ABC] by using the determinants in this 

 as elements. From this definition it is evident that 



[A-BC] = [AB-C] = [ABC], 

 [AB{C + D)] = [ABC] + [ABD], 

 [ABC] = - [ACB] = [CAB]. 



The dot is used throughout our work to show the order of operations. 

 Thus 



[A-BC] = [A[BC]]. 



Products involving the complex line will be considered in §5. 



Since the coordinates ai, a^, as, a.i of a plane can have arbitrary 

 values, not all zero, a linear function of planes is a plane unless it 

 vanishes identically. 



The quantity (ABCD) can be regarded as a product of A and [BCD], 

 of [AB] and [CD], of A, B, C and D etc. Hence by definition and the 

 properties of determinants. 



{ABCD) = {A- BCD) = (AB-CD) = - (ABDC), etc. 



It is to be noted that the sign of (ABCD) is changed when two of the 

 points A, B, C, D are interchanged. These products are called pro- 

 gressive because every additional factor increases the dimension of the 

 product. ' 



If any of these products vanish it shows that the points lie in a 

 space of lower dimension than is determined by a like number of inde- 

 pendent points. For example if 



[ABC] = 



