378 KENNELLY. 



the addition of the imaginary component. It is suggested that the 

 new model here described permits of this reahzation, and thus con- 

 stitutes an embodiment of a complex hyperbolic angle. 



The model only purports to skeletonize part of the upper sheet of 

 three-dimensional surface which it includes. The half beneath the 

 X Y plane is omitted. It seems likely, moreover, that for the pur- 

 pose of analytical completeness, the model should be repeated from O 

 to infinity above the board along the — Y axis. That is, it should 

 probably have the X O Z plane as a plane of symmetry, with a rec- 

 tangular hyperbola at each end of the vertical circle in that plane, 

 and with a series of circles bending away from the vertical on each 

 side of that plane. In the fully developed structure, each of the 

 principal planes X O Y, X O Z, Y O Z would be a plane of symmetry. 

 So far as p^ojecti^'e requirements are concerned, however, this image 

 reduplication of the model about the X Z plane is superfluous. 



Examples in the Use of the Model. 



This model was designed by the writer to facilitate the conception, 

 definition and realization of complex angles, and not for e^'aluating 

 their functions; because the Tables and Charts already publishetl are 

 much more to the purpose of obtaining numerical values. Never- 

 theless, the following simple examples from the Tables may readily 

 be checked by the reader, with reference to Pis. Ill and IV, but to a 

 low degree of precision onlj^: 



cosh ( 1 + iO^) = 0.90700 -|- f0.95076 = 1.31400 Z 46°.349 

 sinh (0.8 - iO^) = 0.62799 - i0.94571 = 1.13522 \ 56°.414 

 cos (0^ + iO.5) = 0.79735 - i'0.36847 = 0.87837 "X 24°.803 

 sin ( + il.2) = + /1. 50946 = 1.50946 Z 90° 



Underscored numerals represent decimal fractions of a circular quad- 

 rant. Thus OJ) = 54°. 



Aeknowledgmenl. 



The writer is indebted to his assistant, Mr. Edy Velander, who, 

 with care and skill, has constructed the model shown in the Plates, 

 and also to his colleague Prof. Louis Derr, for the photographs from 

 which the accompanying Plates have been produced. 



