376 KENNELLY. 



cular component. We then operate with —i on the plane vector so 

 obtained; i. e., we rotate it through 1 quadrant in the X Y plane and 

 in the clockwise direction. An equivalent step is, however, to rotate 

 the X and Y axes of reference in that plane through 1 quadrant in 

 the reverse or positive direction. That is, we may omit the —i 

 operation, if, in dealing with sine projections, we treat O Y as an 

 O X axis, and — O X as an O Y axis, or read off the projections on the 

 X Y plane to the — Y O Y axis as initial line. 



The only difference, therefore, between projecting the cosine and 

 the sine of a complex hyperbolic angle in the model, is that in the 

 latter case the circular component is increased by one quadrant and 

 the projected plane vector is read off to the O Y reference axis as 

 initial line. The model thus gives the projection of either cosh 

 (=1= 01 ± idi) or sinh (±0i ± ie^) within the limits of +1.4 and -1.4 

 for di, and for 02 between the limits + <» and — oo . For accurate 

 numerical work, reference would, of course, be made to the charts 

 and Tables of such functions already published,* and which enable 

 such functions to be obtained either directly or by interpolation, for 

 all ordinary values of di and 62. 



Procedure for Projecting cos (± 0i ='= ido). 



The model enables either the cosine or sine of a hyperbolic complex 

 angle to be projected as a complex quantity or plane vector on the 

 X Y plane. It may also be used for projecting the cosine of a circu- 

 lar complex angle. 



Since cos jS = cosh i^ (6) 



we have, if /3 = ± 0i ± 162, 



cos (± 01 ± 162) = cosh (± idi =F 0o) = cosh (^ 02 =*= ^^i) 



= cosh-(± 02 ^ /0i) = cosh (±0o T /0i) (7) 



In projecting the circular cosine of a complex angle, therefore, we 

 exchange the imaginary and real components, changing the sign of 

 the latter in so doing. We then proceed as though the angle were 

 hyperbolic. The model permits of the projection of cos (=*= 0i =*= idz) 

 between the limits of +00 and — 00 in 0i, and the limits of +1.4 and 

 -1.4 in 02. 



4 Chart Atlas of Complex Hyperbolic and Circular Functions, by A. E. 

 Kennelly, Harvard University Press, 1914. 



