A NEW GEOMETRIC' AL MODEL FOR THE ORTHOGONAL 



PROJECTION OF THE COSINES AND SINES OF 



COMPLEX ANGLES. 



By a. E. Kennelly. 



Received February M, I'.tll). Presented F"ehriiary 12, 1919. 



Geometrical constructions for producing plane-vector representa- 

 tions of cosh (01 + ido), sinh {di + ido), and hence also cos (^i + idz), 

 sin (di + ^d-i), where {di + 162) is a complex argument, or a complex 

 "angle," have been available for some time.^ These constructions 

 involve a rectangular hyperbola and an associated circle, in one and 

 the same plane, which is the plane of the drawing. By making cer- 

 tain projections in this plane, followed by a rotation through a quad- 

 rant, a plane-vector is produced from the origin, corresponding to the 

 complex hyperbolic or circular sine or cosine required. This process 

 is open to the objection that it is somewhat forced and artificial, lack- 

 ing the simple projective property that a sine or cosine of a real angle 

 possesses in either circular or hyperbolic trigonometry. 



More recently, a method of deriving the hyperbolic cosine or sine 

 of a complex angle has been obtained,^ which has enabled the new 

 three-dimensional model here described to be prepared. In this 

 model, it will be seen that the cosine or sine of a complex angle, either 

 hyperbolic or circular, can be produced, by two successive orthogonal 

 projections on to the X Y plane, one projection being made from a 

 rectangular hyperbola, and the other projection being then made 

 from a particular circle definitely selected among a theoretically 

 infinite number of such circles, all concentric at the origin O, which 

 circles, however, are not coplanar. The selection of the particular 

 circle is determined by the foot of the projection from the hyperbola. 

 This effects a geometrical process which is easily apprehended and 



1 " Two Elementary Constructions in Hyperbolic Trigonometry," by A. E. 

 Kennelly, Am. Annals of Mathematics, Salem Pre.ss, 2d Series. Vol. V, 

 No. 4, pp. 181-184, July, 1904, mainly reproduced in "Tables of Complex 

 Hyperbolic and Circular Functions," by A. E. Kennelly, Harvard University 

 Press, 1914, Figs. 19-22, pp. 165-168. 



2 " Artificial Electric Lines," by A. E. Kennelly, McGraw-Hill Book Co., 

 1917. Figs. 68 and 69, pp. 120-121. 



