62, 63] COMPLEX. FLICKER'S COORDINATES. 215 



to itself. The necessary and sufficient condition that a line is self- 

 conjugate is that the pole (focus) of a plane through the line falls 

 in the line. For then as the plane rotates about the line as an axis, 

 the focus describes the line as a ray. Hence the double lines lying 

 in a particular plane all pass through the pole of that plane , and 

 conversely, all the double lines passing through a point lie in the 

 polar plane of the point. Such a system of lines is called by Pliicker 

 a line complex of the first degree. There are in all a double infinity 

 of lines passing through any point in space, but of these only a 

 single infinity belong to the complex. Therefore lines belonging to 

 the complex have one less degrees of freedom than lines in general, 

 or a complex contains a triple infinity of lines. A complex may be 

 represented analytically by a single relation between the four para- 

 meters determining a line. If we mark off on a line any length B, 

 and give its projections on a set of rectangular axes X, Y, Z, and 

 the projections L, M, N of its moment about an origin 0, the line 

 is completely determined. For its direction is given and giving the 

 moment S = T/L 2 -\- M 2 -\- N 2 gives the plane through containing R, 

 and the distance from the line, if the length of R is given, but this 



is given by R = ]/X 2 + Y 2 + Z 2 . 



As the determination of the line is independent of the length 

 of JR, the ratios of the six quantities determine the line. But these 

 five ratios are not independent, for since by 5, 12), 



16) 



we have the identical relation, 



17) LX + MY+ NZ=0, 



expressing the fundamental property that the moment of a vector is 

 perpendicular to it. The coordinates LMNXYZ are known as 

 Pliicker's line -coordinates. 



Thus there remain four independent quantities to determine a 

 line. A relation between these denotes a complex, and in particular 

 a linear relation, 



18) aX + IY + cZ+dL + eM+fN= 0, 



denotes a complex of the first degree. 



Since the double lines of the null- system are the loci of points 

 which are the poles of planes containing the double -lines, at every 

 point of a double -line the resultant moment is perpendicular to it, 



