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Ellery W. Davis 



line their bearer the line, namely, through (x f , y') with slope 

 y"/x". 



(Among the elements of a real line are an infinite number whose 

 imaginary parts are zero. These are the only ones of which we 

 habitually think. 



Scarcely more complicated is the line whose slope is real but 

 whose intercept is complex. Let its equation be 



y = mx -\- b. 

 This breaks into 



\/ = mx' -f- b' and y" - 



mx" + b", 



so that the red vectors join any point on the black line to any 

 point whatsoever on the blue line. The line has no real element 



Fig. 8. 



in the finite region. Moreover, if a vector be an element, the con- 

 jugate vector is not. 



Plainly, if the line joining the black points of two red vectors' 

 is parallel to that joining the blue points, then the vectors belong 

 to one and the same line of real slope. In particular, two red 

 vectors that agree in their black points or in their blue points 

 belong to such a line. 



When blue line coincides with black we get an ordinary 

 real line. 



Consider the more general case where {x x , y x ), (x 2 , y 2 ), call 



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