NP:\V MODKL F(1H OUTIUXJOXAI. PKO.IKCTIOX. 375 



If the real part of the hyperboUc angle is negative, according to 

 the expression cosh ( — ^i =*= ido); then since cosh —(di =f {62) = 

 cosh {9i =F 162), we proceed as in the case of a positive real component, 

 but with a change in the sign of the imaginary component. 



The operation of tracing cosh (=•= 0i =^ id-2) on the X Y plane, thus 

 calls for two successive orthogonal projections onto that plane; 

 namely (1) the projection corresponding to cosh (± ^1) as though 162 

 did not exist, and then (2), the projection corresponding to cosh {62 = 

 cos 62 independently of di, except that the radius of the circle, and its 

 plane, are both conditioned by the magnitude of ^i. 



If we trace the locus of cosh {di ± {$2), where di is held constant, 

 it is evident from an inspection of PI. Ill, that we shall remain on one 

 and the same circle, which projects into one and the same correspond- 

 ing ellipse on the X Y plane. That is, the locus of cosh {di ± 2^2) 

 with di held constant, is an ellipse, whose semi major and minor 

 diameters are cosh di and sinh di respectively. If, on the other hand, 

 we trace cosh (=*=0i + {62) with 62 held constant, we shall run over a 

 certain tie wire bridging all the circles in the model, which tie wire is 

 sin ^2 dm. above the board, and its projection on the board, in the 

 plane X Y of projection, is part of a hyperbola. 



Procedure for sinh (di + {62) 



It would be readily possible to produce a modification of this 

 model here described, which would enable the sine of a complex angle 

 to be projected on the X Y plane following constructions already 

 referred to.^ The transition to a new model for sines is, however, 

 unnecessary. It suffices to use the cosine model here described in a 

 slightly different way. One has only to recall that 



sinh 6 = —i cosh {Q + i-) (4) 



or sinh {di + 162) = -i cosh {^i + {(62 + 5} (5) 



Consequently, in order to find the sine of a complex hyperbolic angle, 

 we proceed on the model as though we sought the cosine of the same 



angle, increased by - radians or 1 quadrant, in the imaginary or cir- 

 3 Artificial Electric Lines, loc. cit. Fig. 69, page 121. 



