374 KENNELLY. 



hyperbolas, tlie foci occupying the already mentioned points +1 and 

 — 1, along — XOX. Such a shadow picture is shown in PI. IV. 



Procedure for Projecting cosh (=<= 0i =*= id^) 



Thus premised, the process of finding the cosine of a complex 

 hyperbolic angle di + 162; that is, the process of fmding cosh {di + ^^2) 

 is as follows : 



Find the arc C E, PI. I, from C = +1 along the rectangular 

 hyperbola C E H, which subtends di radians. The hyperbolic sector 

 comprised between the radius, O C, the hyperbolic arc, and the radius 



vector O E, on this arc from the origin 0, will then include — sq. dm. of 



area. Drop a vertical perpendicular from E on to O X. It will 

 mark off a horizontal distance O D equal to cosh di. Proceed along 

 the circle which rises at D, in a positive or counterclockwise direction ; 

 through 62 circular radians, thus reaching on that circle a point G 

 whose elevation above the drawing board is sin do decimeters. The 

 area enclosed by a radius vector from the origin O on the circle, 



followed between the axis O C and the circular curve, will be — cosh'^i 



2 



sq. dms. 



From G, drop a vertical plummet, as in PI. II, on to the drawing 

 board. In other words, project G orthogonally on the plane X O Y. 

 Let g be the point on the drawing board at which the plummet from 

 G touches the surface. Then it is easily seen that O*; on the drawing 

 board is the required magnitude and direction of cosh {61 -\- 182), in 

 decimeters, with reference to O X as the initial line in the plane X O Y. 

 It may be read off either in rectangular coordinates along axes O X 

 and O Y on a tracing cloth surface as shown in Pi. Ill, or in polar 

 coordinates printed on a sheet seen through the tracing cloth. 



If the circular angle ^2, i- e., the imaginary hyperbolic angle 162, 

 lies between tt and 27r radians, (between 2 and 4 quadrants), the 

 point G will lie on the under side of the plane X Y, and the pro- 

 jection onto g in that plane must be made upwards, instead of down- 

 wards. 



If the hyperbolic angle whose cosine is required has a nogativ^e 

 imaginary component, acconhng to the expression cosh {di — id^), 

 then starting from the projected point I), we must trace out the 

 circular angle in the negative or clockwise direction, as viewed from 

 the front of the model. 



