Ellery W . Davis 



of the ray, the distance between the elements represented is zero. 

 Similarly, the distance between any two elements of a /-ray is 

 zero. 1 If an /-ray and a /-ray have the same center, we have the 

 paradox that although no element of either has any distance from 

 the center, yet the distance of an element of one ray from an 

 element of the other is the square root of the product of the 

 arrow lengths for the elements into the exponential of i times half 

 the angle between them. | 



A very simple construction can be given for the distance from 

 the origin of {x' , iy"), axes being rectangular. Fig. 18 represents 

 the cases 



1 3-/' l< I * I, I y*" I 



Here 



h=\x' + y"\,k=\x'— y" 



> X' 



and c/> = / 1 / 2 = o, o/o, ± n. 



In the first case d\ is the tangent from the origin to the circle of 

 center P' and radius y x " , in the second case d 2 = o; in the third 

 Y 



X 



Fig. i! 



case | d 3 | is the half intercept on the y-axis of the circle of center 

 P' and radius y 3 ". The second case represents a transition from 

 a wholly real to a wholly imaginary distance. 



The distance from one of the three given elements to another of 

 them is wholly imaginary; from (.r', ry/') to (.r', iy 2 ") it is, for 

 example, i{y " — y/'). I" general, to find the distance from any 

 element of a real line to any other element of that line we sub- 

 tract the coordinates of the first element from those of the second 



1 The case where an element is /' or / requires special treatment. 



iS 



