4 Ellery IV. Davis 



turned backward through an angle 6. As 9 varies the black point 

 moves on a circle of radius unity while the blue point moves on 

 one of radius \/2. The red vector is constantly tangent to the 

 inner circle. Had the multiplicand been ( I, — i) the same multi- 

 plier would have caused a turn forward through an angle 6. Had 

 the multiplicand been (a, bi) the figure representing the change 

 would have been simply a parallel projection of figure 4. As 6 

 varied the black point would have moved on the ellipse 



x 2 /a- + y 2 /b 2 = 1 , 



while the blue point moved on 



x 2 /a 2 -\- y 2 /b 2 = 2, 



the red vector keeping tangent to the inner ellipse. The area 

 swept over by the black vector would have to the area of the 

 ellipse the ratio 6 : 2w, i. c, the area would be Bab/ 2. In this more 

 general case, in every position as varies from o to 2tt, the black 

 vector is a semi-diameter of the inner ellipse, while the red vector 

 is parallel and equal to the semi-conjugate diameter. Thus, 

 finally, given any double vector, multiplication by c' 6 will move 

 the black point on an ellipse of which the black vector is a semi- 

 axis, while the red vector is parallel and equal to the semi-conju- 

 gate axis. The red vector will keep tangent to this ellipse, while 

 the black vector sweeps over an area equal to 6 times that of the 

 triangle origin, black point, blue point. The motion will always 

 be such that the red vector is momentarily carried in a direction 

 opposite to that in which we have conceived it to point (fig. 5). 

 Multiplication by re ie , where r is real combines with - the above 

 a simple magnification. 



It is necessary to notice some exceptional cases. 



Suppose that the black vector is collinear with the red. Take 

 for .r-axis the common line and for r-axis the perpendicular 

 thereto. The double vector is then x' -\- ix" = r (cos cf> -f- * sin </>). 

 There are, of course, an infinite number of double vectors having 

 this for the .r-projection. Among them is one whose y-projection 

 is 3'' -)- iy" = — x" -\- ix' =r ( — sin cf> -\- i cos <£) . Thus x' -\- ix" 



