398 



MOORE AND PHILLIPS. 



function of the six edges. These edges can be divided into two sets, 

 those through the point A and those in the plane [BCD]. The Hnes 

 through a point all intersect and so their sum is a line. For the same 

 reason the sum of any number of lines in a plane is a line. Therefore 

 the sum of any number of lines can be expressed as a sum of two lines, 

 one through an arbitrary point A, the other in an arbitrary plane [BCD] 

 not passing through A. 



We have said that a matrix 



V 



Cn ^13 Cii C-2Z C24 C34 



represents a complex line if the elements Cik are not the Pliicker 

 coordinates of a line. It can be expressed as the sum of six matrices 



II ci3 II, etc. 



II C12 II, 



Each of these represents a line. Hence any complex line can be repre- 

 sented as the sum of six lines and consequently as the sum of two lines. 

 Any complex p can then be expressed in the form 



p = [AB] + [CD]. 



By the product of p and any element F (point, line, plane or complex) 

 we mean the sum 



[pT] = [ABT] + [CDT]. 



If p is expressed in a diiferent form 



p = [A'B'] + [CD'] 



it is clear that [pT] will have the same value as before. For example, 

 if r is a point the coordinates of the plane [^SF] + [CDT] are definite 

 linear functions of the sums 



fli bi 

 ttk bk 



+ 



Ci di 

 Ck dk\ 



and the coordinates of [/I'B'F] + [C'D'V] are the same functions of 

 the sums 



a/ 6/ 

 ak bk 



+ 



c/ d,' 

 Ck dk 



Since [AB] + [CD] = [A'B'] + [CD'] those sums are by definition 

 equal. 



