PHILLIPS AND MOORE. — LINEAR DISTANCE AND ANGLE. 



49 



From this definition it is evident that 



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

 [A B{C + D)] = [A BC] + [A B D], 



[ABC] = —[ACB]= [CAB]. 



The sum of any number of three-rowed matrices can be expressed 

 as a single three-rowed matrix [P Q R]. In fact let Ai Bi C\ and 

 A2 Bi Co cut in a line P Q. Then 



[A,B,C,] = [PQR,], 



[A2 Bo Co] = [PQ Ro]. 

 Hence 



Ui By C3] + [.^2 Bo C.] = [PQ Ri] + [PQ Ro] 

 = [PQ {R, + R,)] = [PQ R], 

 where 



R ^ Ri -\- R2. 

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

 determinant 



ttl 02 as 04 



6] ho 63 64 



Ci C2 C3 C4 



di di dz di 



UBCD) = 



This matrix has only one element and hence we write it as a determi- 

 nant. A matrix of one element is analytically equivalent to a number. 

 We use the parentheses to indicate this fact. Square brackets are 

 used to represent matrices which do not reduce to numbers. The 

 vanishing of (A B C D) is the condition that the four points lie in a 

 plane. 



The quantity {A B C D) can be regarded as a product in a number 

 of ways. From the definition it is evident that 



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



4. Regressive Matrices. We can consider space as generated by 

 planes as well as by points. If its coordinates are c^-, a plane a is 

 then represented by a matrix 



a = \\ai ao 03 04 j | . 

 The same plane may be represented by a mq^trix [A B C]. Then the 



