The Imaginary in Geometry 7 



the of the multiplier vary at a finite rate, the projections, upon 

 any line in the finite region, of the end points of the red vector will 

 now one and now the other fly by with infinite velocity. 



We shall usually have no occasion to speak of the black vector, 

 for the red vector and the set of complex coordinates are in one- 

 to-one correspondence. 



The Linear Relation 

 When in the equation of the straight line with real coefficients 

 one of the coordinates is complex, the other will usually also be 

 complex. For example, 



y = mx -f- b, 

 breaks into 



y = mx' -f- b and y" = mx". 



Then the red vectors join any point whatever of the line, say P', 

 to any other point P" . In particular if 



Fig. 7. 



fc5)wm(x> + ix», y + ty'OsP, 



is on the line, so also is the conjugate element, 1 



(*, y) == (x' — ix", y'— fy") =P, 



and is represented by a red vector from 



(.r', y') to (x' — x", y' — y")=P>". 



Conversely, two conjugate vectors (x, y), (x, y) determine a 

 1 Element of a locus will be used to mean that whose coordinates satisfy 

 the equation of the locus. 



