NEW MODEL FOR ORTHOGONAL PROJECTION. 377 



Procedure for Projecting sin (^ 6i ^ id-^. 

 Since sin ,3 = cos (/3 - ^) (8) 



we have by substituting /3 



TTx 



sin (± di ± 702) = COS {(± 01 - ) ± 702} 



= cosh {± 02-t(± 01 - -)} 



= cosh {± 02 + ?'(^ =^ ^i)} (9) 



This procedure diflFers only from that for cos (±0i ='= 702) in altering 

 the imaginary 70i by one quadrant. 



Relations between Complex Hyperbolic and Circular Angles. 



The projective relations above stated for the cosines and sines of 

 both complex hyperbolic and circular angles indicate that while 

 hyperbolic angles relate to rectangular hyperbolic sectors, and cir- 

 cular angles relate to circular sectors, a complex angle relates to an 

 association of a hyperbolic and a circular sector. If the complex 

 angle is hyperbolic, its real part relates to a hyperbolic, and its imagi- 

 nary part to a circular, sector. On the other hand, if the complex 

 angle is circular, its real part relates to a circular, and its imaginary 

 part to a hyperbolic, sector. Complex hyperbolic trigonometry and 

 complex circular trigonometry thus unite in a common geometrical 

 relationship. 



Geometrical Nature of a Complex Hyperbolic Angle. 



In the engineering theory of electric conductors carrying alternat- 

 ing currents, complex hyperbolic angles naturally present themselves.^ 

 The question naturally arises as to how such complex angles may be 

 realized and visualized geometrically. There is no difficulty in the 

 realization of a real h3'perbolic angle. The difficulty only arises with 



5 The Application of Hyperbolic Functions to Electrical Engineering 

 Problems, by A. E. Kennelly, University of London Press, 1912; also Arti- 

 ficial Electric Lines, McGraw-Hill Book Co., 1918. 



