1 6 Ellery IV. Davis 



The most general vector belonging to y = i is {l-\-mi, i), 

 This also, for k larg 

 since .r = I-\- mi gives 



1 



This also, for k large, nearly satisfies y = -g- (.r + k), 



y=[ (l + m i + k)=i(i+-L}- 



I \ m 

 k' 



The black ellipse having a major axis k and a minor axes 

 i -j- I/k at a distance k from the origin coincides in the finite 





I 



\ WW 



_^i^ 



Fig. 16. 



region as £ gets large, with the .r-axes, while the blue ellipse 

 coincides with the lines y 2 = i. The tangents to the black ellipse 

 in the finite region are all drawn from the vertex at (/, o) and 

 are terminated by y=i. The tangent vectors from the vertex 

 ( — ik — /, o) would be terminated by y= — i but would either 

 lie wholly at infinity or be nearly coincident with the .f-axis. 



There are other ways of representing the imaginary element. 

 For example, a vector from the real point on the circular ray 

 through the element and J to the real point on the circular ray 

 through the element and I. This method is due to Cauchy, and 

 has been used by LaGuerre and others. Thus the imaginary 

 element P (fig. 17) on the line AB is represented by the arrow" 

 at right angles to the red vector (x, y) of twice the length of 

 that vector, and bisected by that vector. The conjugate element 

 would be represented by the arrow reversed, that is from the real 

 point on the /-ray to the real point on the /-ray. Since every 

 point in the plane is the center of an /-ray and since that /-ray has 

 the same red vector along AB that the /-ray with its center at the 

 symmetric point has, therefore every point in the plane receives 



16 



