32 Ellery IV. Davis 



By merely revolving the figure we then get all the other supple- 

 mentary curves and their companions. In fig. 28 P 1 and P 1 are 

 elements of the supplementary curve while P and P belong to the 

 companion pair, that is, therefore, to 



£-+;«=(i— «■)■. 



All the types of circles considered have had a real center. It is 

 only necessary to make an imaginary shift to transform them into 

 circles with imaginary centers. For convenience, let the shift of 

 amount il be made parallel to the .r-axis. The circle of complex 

 radius a then becomes 



(x — il) 2 + y 2 = a 2 . 



Every point satisfying x-\-y 2 = a 2 is changed by a red vector il 

 parallel to the .r-axis. As a result of the changes only two points 

 can be real; those namely, if such exist, which satisfy 



,r' 8 4-y a = a' 2 + / 2 — a"* 



— lx' = a'a". 



y 2 =K + / 2 )(i-^), 



and therefore, for a real point, 



I 2 > a"\ 



Fig. 30 illustrates (.r — il) 2 -\- y 2 = 1. 5* and T are the real 

 points, Q and R are the centers of the circular rays through the 

 center O of the circle. 



The completion of the circle x 2 -\-y 2 =\ by the introduction of 

 its complex points enables us to extend our notions of an angle 

 and its functions. The angle 6 = 0' -^-iO" is represented by the 

 black circular sector (fig. 31) of area 6'/2 together with the red 

 hyperbolic sector of area 16/2 whose initial line is the terminal 

 line of the black sector. The functions cos 6 and sin 8 are merely 

 the coordinates of the red vector P. We have, in fact, directly 



32 



These give 



